This problem is same as the previous one, but has larger constraints.

Shiro's just moved to the new house. She wants to invite all friends of her to the house so they can play monopoly. However, her house is too small, so she can only invite one friend at a time.

For each of the n days since the day Shiro moved to the new house, there will be exactly one cat coming to the Shiro's house. The cat coming in the i-th day has a ribbon with color u_i. Shiro wants to know the largest number x, such that if we consider the streak of the first x days, it is possible to remove exactly one day from this streak so that every ribbon color that has appeared among the remaining x - 1 will have the same number of occurrences.

For example, consider the following sequence of u_i: [2, 2, 1, 1, 5, 4, 4, 5]. Then x = 7 makes a streak, since if we remove the leftmost u_i = 5, each ribbon color will appear exactly twice in the prefix of x - 1 days. Note that x = 8 doesn't form a streak, since you must remove exactly one day. 

Since Shiro is just a cat, she is not very good at counting and needs your help finding the longest streak.

Input

The first line contains a single integer n (1 ≤ n ≤ 10^5) — the total number of days.

The second line contains n integers u_1, u_2, …, u_n (1 ≤ u_i ≤ 10^5) — the colors of the ribbons the cats wear. 

Output

Print a single integer x — the largest possible streak of days.

Examples

Input


13
1 1 1 2 2 2 3 3 3 4 4 4 5


Output


13

Input


5
10 100 20 200 1


Output


5

Input


1
100000


Output


1

Input


7
3 2 1 1 4 5 1


Output


6

Input


6
1 1 1 2 2 2


Output


5

Note

In the first example, we can choose the longest streak of 13 days, since upon removing the last day out of the streak, all of the remaining colors 1, 2, 3, and 4 will have the same number of occurrences of 3. Note that the streak can also be 10 days (by removing the 10-th day from this streak) but we are interested in the longest streak.

In the fourth example, if we take the streak of the first 6 days, we can remove the third day from this streak then all of the remaining colors 1, 2, 3, 4 and 5 will occur exactly once.
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.NoSuchElementException;
import java.util.Random;
import java.util.Stack;
import java.util.StringTokenizer;
import java.util.TreeMap;

// CFPS -> CodeForcesProblemSet
public final class CFPS {
	static FastReader fr = new FastReader();
	static PrintWriter out = new PrintWriter(System.out);
	static final int gigamod = 1000000007;
	static int t = 1;
	static double epsilon = 0.00000001;
	
	public static void main(String[] args) {
		OUTER: 
		for (int tc = 0; tc < t; tc++) {
			
			int n = fr.nextInt();
			int[] arr = fr.nextIntArray(n);
			
			// Observations: 
			
			// 1. The task is to find the size of the largest prefix of the
			// array such that every color that appears has the same count
			// except one which has count = 1.
			
			// 2. We will maintain two CountMaps which will work for: 
			// a. Storing counts of all elements.
			// b. Storing counts of counts present.
			
			CountMap elemCounts  = new CountMap();
			CountMap countCounts = new CountMap();
			
			long bestAnswer = 0;
			for (int i = 0; i < n; i++) {
				
				Integer oldElemCnt = elemCounts.get(arr[i]);
				if (oldElemCnt == null)
					oldElemCnt = 0;
				countCounts.remove((int) oldElemCnt);
				elemCounts.put(arr[i]);
				countCounts.put(oldElemCnt + 1);
				
				if (countCounts.size() <= 2) {
					// Checking if we have the solution.
					
					// There are two cases: 
					// a. We completely remove some element.
					// b. We remove an occurence of an element.
					
					Long largestCount = countCounts.lastKey();
					Long smallesCount = countCounts.firstKey();
					
					if (largestCount == smallesCount && elemCounts.size() == 1)
						bestAnswer = i + 1;
					
					if (largestCount == smallesCount && largestCount == 1)
						bestAnswer = i + 1;
					
					if (smallesCount == 1 && countCounts.get(smallesCount) == 1)
						bestAnswer = i + 1;
					
					if (largestCount - smallesCount != 1) continue;
					
					if (countCounts.get(largestCount) == 1)
						bestAnswer = i + 1;
				}
			}
			
			out.println(bestAnswer);
		}
		out.close();
	}
	
	static long power(long x, int y)
	  {
		// int p = 998244353;
		// int p = gigamod;
	    long res = 1; // Initialize result
	 
	    // x = x % p; // Update x if it is more than or
	    // equal to p
	 
	    if (x == 0)
	      return 0; // In case x is divisible by p;
	 
	    while (y > 0)
	    {
	 
	      // If y is odd, multiply x with result
	      if ((y & 1) != 0)
	        res = (res * x)/* % p*/;
	 
	      // y must be even now
	      y = y >> 1; // y = y/2
	      x = (x * x)/* % p*/;
	    }
	    return res;
	  }
	
	// Maps elements in a 2D matrix serially to elements in 
	// a 1D array.
	static int mapTo1D(int row, int col, int n, int m) {
		return row * m + col;
	}
	
	// Inverse of what the one above does.
	static int[] mapTo2D(int idx, int n, int m) {
		int[] rnc = new int[2];
		rnc[0] = idx / m;
		rnc[1] = idx % m;
		return rnc;
	}
	
	// Checks if s has subsequence t.
	static boolean hasSubsequence(String s, String t) {
		char[] schars = s.toCharArray();
		char[] tchars = t.toCharArray();
		int slen = schars.length, tlen = tchars.length;
		int tctr = 0;
		
		if (slen < tlen) return false;
		
		for (int i = 0; i < slen || i < tlen; i++) {
			if (tctr == tlen) break;
			if (schars[i] == tchars[tctr]) {
				tctr++;
			}
		}
		
		if (tctr == tlen) return true;
		return false;
	}
	
	// Returns the binary string of length at least bits.
	static String toBinaryString(long num, int bits) {
		StringBuilder sb = new StringBuilder(Long.toBinaryString(num));
		sb.reverse();
		for (int i = sb.length(); i < bits; i++)
			sb.append('0');
		return sb.reverse().toString();
	}
	
	static class CountMap extends TreeMap<Long, Integer>{
		
		CountMap() {
		}
		
		CountMap(int[] arr) {
			put(arr);
		}
		
		CountMap(long[] arr) {
			put(arr);
		}
		
		CountMap(CountMap cm) {
			
		}
		
		public void removeTM(Long key) {
			super.remove(key);
		}
		
		public void removeTM(Integer key) {
			super.remove((long) key);
		}
		
		public Integer put(Long key) {
			if (super.containsKey(key)) {
				return super.put(key, super.get(key) + 1);
			} else {
				return super.put(key, 1);
			}
		}
		
		public Integer put(int key) {
			if (super.containsKey((long) key)) {
				return super.put((long) key, super.get((long) key) + 1);
			} else {
				return super.put((long) key, 1);
			}
		}
		
		public Integer remove(Long key) {
			Integer count = super.get(key);
			if (count == null) return -1;
			if (count == 1)
				return super.remove(key);
			else
				return super.put(key, super.get(key) - 1);
		}
		
		public Integer remove(int key) {
			Integer count = super.get((long) key);
			if (count == null) return -1;
			if (count == 1)
				return super.remove((long) key);
			else
				return super.put((long) key, super.get((long) key) - 1);
		}
		
		public Integer get(int key) {
			Integer count = super.get((long) key);
			if (count == null)
				return 0;
			return count;
		}
		
		public Integer get(long key) {
			Integer count = super.get(key);
			if (count == null)
				return 0;
			return count;
		}
	
		public void put(int[] arr) {
			for (int i : arr)
				this.put(i);
		}
		
		public void put(long[] arr) {
			for (long l : arr)
				this.put(l);
		}
	}
	
	static class Point implements Comparable<Point> {
		long x;
		long y;
		long id;
		
		Point() {
			x = y = id = 0;
		}
		
		Point(Point p) {
			this.x = p.x;
			this.y = p.y;
			this.id = p.id;
		}
		
		Point(long a, long b, long id) {
			this.x = a;
			this.y = b;
			this.id = id;
		}
		
		Point(long a, long b) {
			this.x = a;
			this.y = b;
		}
		
		@Override
		public int compareTo(Point o) {
			if (this.x > o.x)
				return 1;
			if (this.x < o.x)
				return -1;
			if (this.y > o.y)
				return 1;
			if (this.y < o.y)
				return -1;
			return 0;
		}
		
		public boolean equals(Point that) {
			return this.compareTo(that) == 0;
		}
	}
	
	static class PointComparator implements Comparator<Point> {
		@Override
		public int compare(Point o1, Point o2) {
			
			// Comparision will be done on the basis of the start
			// of the segment and then on the length of the segment.
			
			if (o1.x > o2.x) return 1;
			if (o1.x < o2.x) return -1;
			
			if (o1.y - o1.x > o2.y - o2.x) return 1;
			if (o1.y - o1.x < o2.y - o2.x) return -1;
			
			return 0;
		}
	}
	
	static double distToOrigin(Point p1) {
		return Math.sqrt(Math.pow(p1.y, 2) + Math.pow(p1.x, 2));
	}
	
	static double distance(Point p1, Point p2) {
		
		long y2 = p2.y, y1 = p1.y;
		long x2 = p2.x, x1 = p1.x;
		
		double distance = Math.sqrt(Math.pow(y2-y1, 2) + Math.pow(x2-x1, 2));
		return distance;
	}
	
	// Returns the largest power of k that fits into n.
	static int largestFittingPower(long n, long k) {
		int lo = 0, hi = logk(Long.MAX_VALUE, 3);
		int largestPower = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			
			long val = (long) Math.pow(k, mid);
			if (val <= n) {
				largestPower = mid;
				lo = mid + 1;
			} else {
				hi = mid - 1;
			}
		}
		return largestPower;
	}
	
	// Returns map of factor and its power in the number.
	static HashMap<Long, Integer> primeFactorization(long num) {
		HashMap<Long, Integer> map = new HashMap<>();
		while (num % 2 == 0) {
			num /= 2;
			Integer pwrCnt = map.get(2L);
			map.put(2L, pwrCnt != null ? pwrCnt + 1 : 1);
		}
		
		for (long i = 3; i * i <= num; i += 2) {
			while (num % i == 0) {
				num /= i;
				Integer pwrCnt = map.get(i);
				map.put(i, pwrCnt != null ? pwrCnt + 1 : 1);
			}
		}
		
		// If the number is prime, we have to add it to the 
		// map.
		if (num != 1)
			map.put(num, 1);
		
		return map;
	}
	
	// Returns map of factor and its power in the number.
	static TreeMap<Integer, Integer> primeFactorization(int num) {
		TreeMap<Integer, Integer> map = new TreeMap<>();
		while (num % 2 == 0) {
			num /= 2;
			Integer pwrCnt = map.get(2);
			map.put(2, pwrCnt != null ? pwrCnt + 1 : 1);
		}
		
		for (int i = 3; i * i <= num; i += 2) {
			while (num % i == 0) {
				num /= i;
				Integer pwrCnt = map.get(i);
				map.put(i, pwrCnt != null ? pwrCnt + 1 : 1);
			}
		}
		
		// If the number is prime, we have to add it to the 
		// map.
		if (num != 1)
			map.put(num, 1);
		
		return map;
	}
	
	static HashSet<Long> divisors(long num) {
		HashSet<Long> divisors = new HashSet<Long>();
		divisors.add(1L);
		divisors.add(num);
		
		for (long i = 2; i * i <= num; i++) {
			if (num % i == 0) {
				divisors.add(num/i);
				divisors.add(i);
			}
		}
		return divisors;
	}
	
	// Returns the index of the first element
	// larger than or equal to val.
	static int bsearch(int[] arr, int val, int lo, int hi) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] >= val) {
				idx = mid;
				hi = mid - 1;
			} else
				lo = mid + 1;
		}
		return idx;
	}
	
	static int bsearch(long[] arr, long val, int lo, int hi) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] >= val) {
				idx = mid;
				hi = mid - 1;
			} else
				lo = mid + 1;
		}
		return idx;
	}
	
	// Returns the index of the last element 
	// smaller than or equal to val.
	static int bsearch(long[] arr, long val, int lo, int hi, boolean sMode) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] > val) {
				hi = mid - 1;
			} else {
				idx = mid;
				lo = mid + 1;
			}
		}
		return idx;
	}
	
	static int bsearch(int[] arr, long val, int lo, int hi, boolean sMode) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] > val) {
				hi = mid - 1;
			} else {
				idx = mid;
				lo = mid + 1;
			}
		}
		return idx;
	}
	
	static long factorial(long n) {
		if (n <= 1)
			return 1;
		long factorial = 1;
		for (int i = 1; i <= n; i++)
			factorial = mod(factorial * i);
		return factorial;
	}
	
	static long factorialInDivision(long a, long b) {
		if (a == b)
			return 1;
		if (b < a) {
			long temp = a;
			a = b;
			b = temp;
		}
		
		long factorial = 1;
		for (long i = a + 1; i <= b; i++)
			factorial = mod(factorial * i);
		return factorial;
	}
	
	static BigInteger factorialInDivision(BigInteger a, BigInteger b) {
		if (a.equals(b))
			return BigInteger.ONE;
		return a.multiply(factorialInDivision(a.subtract(BigInteger.ONE), b));
	}
	
	static long nCr(long n, long r) {
		long p = gigamod;
	    // Base case
	    if (r == 0)
	        return 1;
	 
	    // Fill factorial array so that we
	    // can find all factorial of r, n
	    // and n-r
	    long fac[] = new long[(int)n + 1];
	    fac[0] = 1;
	    for (int i = 1; i <= n; i++)
	        fac[i] = fac[i - 1] * i % p;
	 
	    return (fac[(int)n] * modInverse(fac[(int)r], p) % p
	                    * modInverse(fac[(int)n - (int)r], p) % p) % p;
	}
	
	static long modInverse(long n, long p) {
	    return power(n, p - 2, p);
	}
	
	static long power(long x, long y, long p) {
	    long res = 1; // Initialize result
	 
	    x = x % p; // Update x if it is more than or
	    // equal to p
	 
	    while (y > 0) {
	        // If y is odd, multiply x with result
	        if ((y & 1)==1)
	            res = (res * x) % p;
	 
	        // y must be even now
	        y = y >> 1; // y = y/2
	        x = (x * x) % p;
	    }
	    return res;
	}
	
	static long nPr(long n, long r) {
		return factorialInDivision(n, n - r);
	}
	
	static int log2(long n) {
		return (int)(Math.log(n) / Math.log(2));
	}
	
	static double log2(long n, boolean doubleMode) {
		return (Math.log(n) / Math.log(2));
	}
	
	static int logk(long n, long k) {
		return (int)(Math.log(n) / Math.log(k));
	}
	
	// Sieve of Eratosthenes: 
	static boolean[] primeGenerator(int upto) {
		boolean[] isPrime = new boolean[upto + 1];
		Arrays.fill(isPrime, true);
		isPrime[1] = isPrime[0] = false;

		for (long i = 2; i * i < upto + 1; i++)
			if (isPrime[(int) i])
				// Mark all the multiples greater than or equal
				// to the square of i to be false.
				for (long j = i; j * i < upto + 1; j++)
					isPrime[(int) j * (int) i] = false;
		return isPrime;
	}

	static int gcd(int a, int b) {
		if (b == 0) {
			return a;
		} else {
			return gcd(b, a % b);
		}
	}
	
	static long gcd(long a, long b) {
		if (b == 0) {
			return a;
		} else {
			return gcd(b, a % b);
		}
	}
	
	static int gcd(int[] arr) {
		int n = arr.length;
		int gcd = arr[0];
		for (int i = 1; i < n; i++) {
			gcd = gcd(gcd, arr[i]);
		}
		return gcd;
	}
	
	static long gcd(long[] arr) {
		int n = arr.length;
		long gcd = arr[0];
		for (int i = 1; i < n; i++) {
			gcd = gcd(gcd, arr[i]);
		}
		return gcd;
	}
	
	static long lcm(int[] arr) {
		long lcm = arr[0];
		int n = arr.length;
		for (int i = 1; i < n; i++) {
			lcm = (lcm * arr[i]) / gcd(lcm, arr[i]);
		}
		return lcm;
	}
	
	static long lcm(long[] arr) {
		long lcm = arr[0];
		int n = arr.length;
		for (int i = 1; i < n; i++) {
			lcm = (lcm * arr[i]) / gcd(lcm, arr[i]);
		}
		return lcm;
	}
	
	static long lcm(int a, int b) {
		return (a * b)/gcd(a, b);
	}
	
	static long lcm(long a, long b) {
		return (a * b)/gcd(a, b);
	}
	
	static boolean less(int a, int b) {
		return a < b ? true : false;
	}
	
	static boolean isSorted(int[] a) {
		for (int i = 1; i < a.length; i++) {
			if (less(a[i], a[i - 1]))
				return false;
		}
		return true;
	}
	
	static boolean isSorted(long[] a) {
		for (int i = 1; i < a.length; i++) {
			if (a[i] < a[i - 1])
				return false;
		}
		return true;
	}

	static void swap(int[] a, int i, int j) {
		int temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(long[] a, int i, int j) {
		long temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(double[] a, int i, int j) {
		double temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(char[] a, int i, int j) {
		char temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void sort(int[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(char[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(long[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(double[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void reverseSort(int[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(char[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(long[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(double[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void shuffleArray(long[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			long tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static void shuffleArray(int[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			int tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static void shuffleArray(double[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			double tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	private static void shuffleArray(char[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			char tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static boolean isPrime(long n) {
		if (n <= 1)
			return false;
		if (n <= 3)
			return true;

		if (n % 2 == 0 || n % 3 == 0)
			return false;

		for (long i = 5; i * i <= n; i = i + 6)
			if (n % i == 0 || n % (i + 2) == 0)
				return false;

		return true;
	}

	static String toString(int[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(boolean[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(long[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(char[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + "");
		return sb.toString();
	}
	
	static String toString(int[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + "");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(long[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(double[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(char[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static char toChar(int i) {
		return (char) (i + 48);
	}
	
	static long mod(long a, long m) {
	    return (a%m + m) % m;
	}
	
	static long mod(long num) {
		return (num % gigamod + gigamod) % gigamod;
	}
	
	// Uses weighted quick-union with path compression.
	static class UnionFind {
	    private int[] parent;  // parent[i] = parent of i
	    private int[] size;    // size[i] = number of sites in tree rooted at i
	                           // Note: not necessarily correct if i is not a root node
	    private int count;     // number of components

	    public UnionFind(int n) {
	        count = n;
	        parent = new int[n];
	        size = new int[n];
	        for (int i = 0; i < n; i++) {
	            parent[i] = i;
	            size[i] = 1;
	        }
	    }

	    // Number of connected components.
	    public int count() {
	        return count;
	    }
	  
	    // Find the root of p.
	    public int find(int p) {
	        int root = p;
	        while (root != parent[root])
	            root = parent[root];
	        while (p != root) {
	            int newp = parent[p];
	            parent[p] = root;
	            p = newp;
	        }
	        return root;
	    }

	    public boolean connected(int p, int q) {
	        return find(p) == find(q);
	    }
	    
	    public int numConnectedTo(int node) {
	    	return size[find(node)];
	    }
	    
	    // Weighted union.
	    public void union(int p, int q) {
	        int rootP = find(p);
	        int rootQ = find(q);
	        
	        if (rootP == rootQ) return;

	        // make smaller root point to larger one
	        if (size[rootP] < size[rootQ]) {
	            parent[rootP] = rootQ;
	            size[rootQ] += size[rootP];
	        }
	        else {
	            parent[rootQ] = rootP;
	            size[rootP] += size[rootQ];
	        }
	        count--;
	    }
	    
	    public static int[] connectedComponents(UnionFind uf) {
	    	// We can do this in nlogn.
	    	
	    	int n = uf.size.length;
	    	int[] compoColors = new int[n];
	    	for (int i = 0; i < n; i++)
	    		compoColors[i] = uf.find(i);
	    	
	    	HashMap<Integer, Integer> oldToNew = new HashMap<>();
	    	int newCtr = 0;
	    	
	    	for (int i = 0; i < n; i++) {
	    		
	    		int thisOldColor = compoColors[i];
	    		Integer thisNewColor = oldToNew.get(thisOldColor);
	    		if (thisNewColor == null)
	    			thisNewColor = newCtr++;
	    		
	    		oldToNew.put(thisOldColor, thisNewColor);
	    		compoColors[i] = thisNewColor;
	    	}
	    	
	    	return compoColors;
	    }
	}
	
	static class UGraph {
		// Adjacency list.
		private HashSet<Integer>[] adj;
		private static final String NEWLINE = "\n";
		private int E;
		
		public UGraph(int V) {
			adj = (HashSet<Integer>[]) new HashSet[V];
			E = 0;
			for (int i = 0; i < V; i++)
				adj[i] = new HashSet<Integer>();
		}
		
		public void addEdge(int from, int to) {
			if (adj[from].contains(to)) return;
			E++;
			adj[from].add(to);
			adj[to].add(from);
		}
		
		public HashSet<Integer> adj(int from) {
			return adj[from];
		}
		
		public int degree(int v) {
			return adj[v].size();
		}
		
		public int V() {
			return adj.length;
		}
		
		public int E() {
			return E;
		}
		
		public String toString() {
			StringBuilder s = new StringBuilder();
			s.append(V() + " vertices, " + E() + " edges " + NEWLINE);
			for (int v = 0; v < V(); v++) {
				s.append(v + ": ");
				for (int w : adj[v]) {
					s.append(w + " ");
				}
				s.append(NEWLINE);
			}
			return s.toString();
		}
		
		public static void dfsMark(int current, boolean[] marked, UGraph g) {
			if (marked[current]) return;
			
			marked[current] = true;
			Iterable<Integer> adj = g.adj(current);
			for (int adjc : adj)
				dfsMark(adjc, marked, g);
		}
		
		public static void dfsMark(int current, int from, long[] distTo, boolean[] marked, UGraph g, ArrayList<Integer> endPoints) {
			if (marked[current]) return;
			
			marked[current] = true;
			if (from != -1)
				distTo[current] = distTo[from] + 1;
			HashSet<Integer> adj = g.adj(current);
			int alreadyMarkedCtr = 0;
			for (int adjc : adj) {
				if (marked[adjc]) alreadyMarkedCtr++;
				dfsMark(adjc, current, distTo, marked, g, endPoints);
			}
			
			if (alreadyMarkedCtr == adj.size())
				endPoints.add(current);
		}
		
		public static void bfsOrder(int current, UGraph g) {
		}
		
		public static void dfsMark(int current, int[] colorIds, int color, UGraph g) {
			if (colorIds[current] != -1) return;
			
			colorIds[current] = color;
			Iterable<Integer> adj = g.adj(current);
			for (int adjc : adj)
				dfsMark(adjc, colorIds, color, g);
		}
		
		public static int[] connectedComponents(UGraph g) {
			int n = g.V();
			int[] componentId = new int[n];
			Arrays.fill(componentId, -1);
			int colorCtr = 0;
			for (int i = 0; i < n; i++) {
				if (componentId[i] != -1) continue;
				dfsMark(i, componentId, colorCtr, g);
				colorCtr++;
			}
			
			return componentId;
		}
		
		public static boolean hasCycle(UGraph ug) {
			int n = ug.V();
			boolean[] marked = new boolean[n];
			boolean[] hasCycleFirst = new boolean[1];
			for (int i = 0; i < n; i++) {
				if (marked[i]) continue;
				hcDfsMark(i, ug, marked, hasCycleFirst, -1);
			}
			return hasCycleFirst[0];
		}
		
		// Helper for hasCycle.
		private static void hcDfsMark(int current, UGraph ug, boolean[] marked, boolean[] hasCycleFirst, int parent) {
			if (marked[current]) return;
			if (hasCycleFirst[0]) return;
			
			marked[current] = true;
			HashSet<Integer> adjc = ug.adj(current);
			for (int adj : adjc) {
				if (marked[adj] && adj != parent && parent != -1) {
					hasCycleFirst[0] = true;
					return;
				}
				
				hcDfsMark(adj, ug, marked, hasCycleFirst, current);
			}
		}
	}
	
	static class Digraph {
		// Adjacency list.
		private HashSet<Integer>[] adj;
		private static final String NEWLINE = "\n";
		private int E;
		
		public Digraph(int V) {
			adj = (HashSet<Integer>[]) new HashSet[V];
			E = 0;
			for (int i = 0; i < V; i++)
				adj[i] = new HashSet<Integer>();
		}
		
		public void addEdge(int from, int to) {
			if (adj[from].contains(to)) return;
			E++;
			adj[from].add(to);
		}
		
		public HashSet<Integer> adj(int from) {
			return adj[from];
		}
		
		public int V() {
			return adj.length;
		}
		
		public int E() {
			return E;
		}
		
		public String toString() {
			StringBuilder s = new StringBuilder();
			s.append(V() + " vertices, " + E() + " edges " + NEWLINE);
			for (int v = 0; v < V(); v++) {
				s.append(v + ": ");
				for (int w : adj[v]) {
					s.append(w + " ");
				}
				s.append(NEWLINE);
			}
			return s.toString();
		}
		
		public static void dfsMark(int source, boolean[] marked, Digraph g) {
			if (marked[source]) return;
			
			marked[source] = true;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, marked, g);
		}
		
		public static void bfsOrder(int source, Digraph g) {
		}
		
		private static void dfsMark(int source, int[] colorIds, int color, Digraph g) {
			if (colorIds[source] != -1) return;
			
			colorIds[source] = color;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, colorIds, color, g);
		}
		
		public static int[] connectedComponents(Digraph g) {
			int n = g.V();
			int[] componentId = new int[n];
			Arrays.fill(componentId, -1);
			int colorCtr = 0;
			for (int i = 0; i < n; i++) {
				if (componentId[i] != -1) continue;
				dfsMark(i, componentId, colorCtr, g);
				colorCtr++;
			}
			
			return componentId;
		}
		
		public static Stack<Integer> topologicalSort(Digraph dg)
		{
			// dg has to be a directed acyclic graph.
			// We'll have to run dfs on the digraph and push the deepest nodes on stack first.
			// We'll need a Stack<Integer> and a int[] marked.
			Stack<Integer> topologicalStack = new Stack<Integer>();
			boolean[] marked = new boolean[dg.V()];
			
			// Calling dfs
			for (int i = 0; i < dg.V(); i++)
			{
				if (!marked[i]) runDfs(dg, topologicalStack, marked, i);
			}
			
			return topologicalStack;
		}
		
		static void runDfs(Digraph dg, Stack<Integer> topologicalStack, boolean[] marked, int source)
		{
			marked[source] = true;
			for (Integer adjVertex : dg.adj(source))
			{
				if (!marked[adjVertex]) runDfs(dg, topologicalStack, marked, adjVertex);
			}
			topologicalStack.add(source);
		}
	}
	
	public static class DirectedEdge implements Comparable<DirectedEdge> { 
	    private final int v;
	    private final int w;
	    private final double weight;

	    public DirectedEdge(int v, int w, double weight) {
	        this.v = v;
	        this.w = w;
	        this.weight = weight;
	    }

	    public int from() {
	        return v;
	    }

	    public int to() {
	        return w;
	    }
	    
	    public double weight() {
	        return weight;
	    }
	    
	    public String toString() {
	        return v + "->" + w + " " + String.format("%5.2f", weight);
	    }

		@Override
		public int compareTo(DirectedEdge o) {
			if (this.v > o.v)
				return 1;
			if (this.v < o.v)
				return -1;
			if (this.w > o.w)
				return 1;
			if (this.w < o.w)
				return -1;
			if (this.weight > o.weight)
				return 1;
			if (this.weight < o.weight)
				return -1;
			return 0;
		}
	}
	
	public static class Edge implements Comparable<Edge> { 

	    private final int v;
	    private final int w;
	    private final double weight;

	    public Edge(int v, int w, double weight) {
	        if (v < 0) throw new IllegalArgumentException("vertex index must be a nonnegative integer");
	        if (w < 0) throw new IllegalArgumentException("vertex index must be a nonnegative integer");
	        if (Double.isNaN(weight)) throw new IllegalArgumentException("Weight is NaN");
	        this.v = v;
	        this.w = w;
	        this.weight = weight;
	    }

	    public double weight() {
	        return weight;
	    }

	    public int either() {
	        return v;
	    }

	    public int other(int vertex) {
	        if      (vertex == v) return w;
	        else if (vertex == w) return v;
	        else throw new IllegalArgumentException("Illegal endpoint");
	    }

	    @Override
	    public int compareTo(Edge that) {
	        return Double.compare(this.weight, that.weight);
	    }

	    public String toString() {
	        return String.format("%d-%d %.5f", v, w, weight);
	    }
	}
	
	public static class EdgeWeightedDigraph {
	    private static final String NEWLINE = System.getProperty("line.separator");

	    private final int V;                // number of vertices in this digraph
	    private int E;                      // number of edges in this digraph
	    private HashSet<DirectedEdge>[] adj;    // adj[v] = adjacency list for vertex v
	    private int[] indegree;             // indegree[v] = indegree of vertex v
	    
	    public EdgeWeightedDigraph(int V) {
	        if (V < 0) throw new IllegalArgumentException("Number of vertices in a Digraph must be nonnegative");
	        this.V = V;
	        this.E = 0;
	        this.indegree = new int[V];
	        adj = (HashSet<DirectedEdge>[]) new HashSet[V];
	        for (int v = 0; v < V; v++)
	            adj[v] = new HashSet<DirectedEdge>();
	    }

	    /**
	     * Initializes a new edge-weighted digraph that is a deep copy of {@code G}.
	     *
	     * @param  G the edge-weighted digraph to copy
	     */
	    public EdgeWeightedDigraph(EdgeWeightedDigraph G) {
	        this(G.V());
	        this.E = G.E();
	        for (int v = 0; v < G.V(); v++)
	            this.indegree[v] = G.indegree(v);
	        for (int v = 0; v < G.V(); v++) {
	            // reverse so that adjacency list is in same order as original
	            Stack<DirectedEdge> reverse = new Stack<DirectedEdge>();
	            for (DirectedEdge e : G.adj[v]) {
	                reverse.push(e);
	            }
	            for (DirectedEdge e : reverse) {
	                adj[v].add(e);
	            }
	        }
	    }

	    public int V() {
	        return V;
	    }

	    public int E() {
	        return E;
	    }

	    public void addEdge(DirectedEdge e) {
	        int v = e.from();
	        int w = e.to();
	        adj[v].add(e);
	        indegree[w]++;
	        E++;
	    }
	    
	    public void addEdge(int from, int to, long weight) {
	    	addEdge(new DirectedEdge(from, to, weight));
	    }

	    public Iterable<DirectedEdge> adj(int v) {
	        return adj[v];
	    }

	    public int outdegree(int v) {
	        return adj[v].size();
	    }
	    
	    public int indegree(int v) {
	        return indegree[v];
	    }
	    
	    public Iterable<DirectedEdge> edges() {
	        HashSet<DirectedEdge> list = new HashSet<DirectedEdge>();
	        for (int v = 0; v < V; v++) {
	            for (DirectedEdge e : adj(v)) {
	                list.add(e);
	            }
	        }
	        return list;
	    } 

	    public String toString() {
	        StringBuilder s = new StringBuilder();
	        s.append(V + " " + E + NEWLINE);
	        for (int v = 0; v < V; v++) {
	            s.append(v + ": ");
	            for (DirectedEdge e : adj[v]) {
	                s.append(e + "  ");
	            }
	            s.append(NEWLINE);
	        }
	        return s.toString();
	    }
	}
	
	public static class EdgeWeightedGraph {
	    private static final String NEWLINE = System.getProperty("line.separator");

	    private final int V;
	    private int E;
	    private HashSet<Edge>[] adj;
	    
	    public EdgeWeightedGraph(int V) {
	        this.V = V;
	        this.E = 0;
	        adj = (HashSet<Edge>[]) new HashSet[V];
	        for (int v = 0; v < V; v++) {
	            adj[v] = new HashSet<Edge>();
	        }
	    }

	    public EdgeWeightedGraph(EdgeWeightedGraph G) {
	        this(G.V());
	        this.E = G.E();
	        for (int v = 0; v < G.V(); v++) {
	            // reverse so that adjacency list is in same order as original
	            Stack<Edge> reverse = new Stack<Edge>();
	            for (Edge e : G.adj[v]) {
	                reverse.push(e);
	            }
	            for (Edge e : reverse) {
	                adj[v].add(e);
	            }
	        }
	    }

	    public int V() {
	        return V;
	    }

	    public int E() {
	        return E;
	    }

	    public void addEdge(Edge e) {
	        int v = e.either();
	        int w = e.other(v);
	        adj[v].add(e);
	        adj[w].add(e);
	        E++;
	    }
	    
	    public void addEdge(int from, int to, double weight) {
	    	addEdge(new Edge(from, to, weight));
	    }

	    public Iterable<Edge> adj(int v) {
	        return adj[v];
	    }

	    public int degree(int v) {
	        return adj[v].size();
	    }

	    public Iterable<Edge> edges() {
	        HashSet<Edge> list = new HashSet<Edge>();
	        for (int v = 0; v < V; v++) {
	            int selfLoops = 0;
	            for (Edge e : adj(v)) {
	                if (e.other(v) > v) {
	                    list.add(e);
	                }
	                // add only one copy of each self loop (self loops will be consecutive)
	                else if (e.other(v) == v) {
	                    if (selfLoops % 2 == 0) list.add(e);
	                    selfLoops++;
	                }
	            }
	        }
	        return list;
	    }
	    
	    public String toString() {
	        StringBuilder s = new StringBuilder();
	        s.append(V + " " + E + NEWLINE);
	        for (int v = 0; v < V; v++) {
	            s.append(v + ": ");
	            for (Edge e : adj[v]) {
	                s.append(e + "  ");
	            }
	            s.append(NEWLINE);
	        }
	        return s.toString();
	    }
	}
	
	static class FastReader {
		private BufferedReader bfr;
		private StringTokenizer st;
		public FastReader() {
			bfr = new BufferedReader(new InputStreamReader(System.in));
		}

		String next() {
			if (st == null || !st.hasMoreElements()) {
				try {
					st = new StringTokenizer(bfr.readLine());
				} catch (IOException e) {
					e.printStackTrace();
				}
			}
			return st.nextToken();
		}

		int nextInt() {
			return Integer.parseInt(next());
		}

		long nextLong() {
			return Long.parseLong(next());
		}

		double nextDouble() {
			return Double.parseDouble(next());
		}

		char nextChar() {
			return next().toCharArray()[0];
		}

		String nextString() {
			return next();
		}

		int[] nextIntArray(int n) {
			int[] arr = new int[n];
			for (int i = 0; i < n; i++)
				arr[i] = nextInt();
			return arr;
		}

		double[] nextDoubleArray(int n) {
			double[] arr = new double[n];
			for (int i = 0; i < arr.length; i++)
				arr[i] = nextDouble();
			return arr;
		}

		long[] nextLongArray(int n) {
			long[] arr = new long[n];
			for (int i = 0; i < n; i++)
				arr[i] = nextLong();
			return arr;
		}
	}
	
	private static class IndexMaxPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
	    private int maxN;        // maximum number of elements on PQ
	    private int n;           // number of elements on PQ
	    private int[] pq;        // binary heap using 1-based indexing
	    private int[] qp;        // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
	    private Key[] keys;      // keys[i] = priority of i

	    public IndexMaxPQ(int maxN) {
	        if (maxN < 0) throw new IllegalArgumentException();
	        this.maxN = maxN;
	        n = 0;
	        keys = (Key[]) new Comparable[maxN + 1];    // make this of length maxN??
	        pq   = new int[maxN + 1];
	        qp   = new int[maxN + 1];                   // make this of length maxN??
	        for (int i = 0; i <= maxN; i++)
	            qp[i] = -1;
	    }

	    public boolean isEmpty() {
	        return n == 0;
	    }

	    public boolean contains(int i) {
	        validateIndex(i);
	        return qp[i] != -1;
	    }

	    public int size() {
	        return n;
	    }

	   /**
	     * Associate key with index i.
	     *
	     * @param  i an index
	     * @param  key the key to associate with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if there already is an item
	     *         associated with index {@code i}
	     */
	    public void insert(int i, Key key) {
	        validateIndex(i);
	        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
	        n++;
	        qp[i] = n;
	        pq[n] = i;
	        keys[i] = key;
	        swim(n);
	    }

	    /**
	     * Returns an index associated with a maximum key.
	     *
	     * @return an index associated with a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int maxIndex() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return pq[1];
	    }

	    /**
	     * Returns a maximum key.
	     *
	     * @return a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public Key maxKey() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return keys[pq[1]];
	    }

	    /**
	     * Removes a maximum key and returns its associated index.
	     *
	     * @return an index associated with a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int delMax() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        int max = pq[1];
	        exch(1, n--);
	        sink(1);

	        assert pq[n+1] == max;
	        qp[max] = -1;        // delete
	        keys[max] = null;    // to help with garHashSete collection
	        pq[n+1] = -1;        // not needed
	        return max;
	    }

	    /**
	     * Returns the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to return
	     * @return the key associated with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public Key keyOf(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        else return keys[i];
	    }

	    /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     */
	    public void changeKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	        sink(qp[i]);
	    }

	    /**
	     * Increase the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to increase
	     * @param  key increase the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key <= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void increaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) > 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key that is strictly less than the key in the priority queue");

	        keys[i] = key;
	        swim(qp[i]);
	    }

	    /**
	     * Decrease the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to decrease
	     * @param  key decrease the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key >= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void decreaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) < 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key that is strictly greater than the key in the priority queue");
	        keys[i] = key;
	        sink(qp[i]);
	    }

	    /**
	     * Remove the key on the priority queue associated with index {@code i}.
	     *
	     * @param  i the index of the key to remove
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void delete(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        int index = qp[i];
	        exch(index, n--);
	        swim(index);
	        sink(index);
	        keys[i] = null;
	        qp[i] = -1;
	    }

	    // throw an IllegalArgumentException if i is an invalid index
	    private void validateIndex(int i) {
	        if (i < 0) throw new IllegalArgumentException("index is negative: " + i);
	        if (i >= maxN) throw new IllegalArgumentException("index >= capacity: " + i);
	    }
	    
	    private boolean less(int i, int j) {
	        return keys[pq[i]].compareTo(keys[pq[j]]) < 0;
	    }

	    private void exch(int i, int j) {
	        int swap = pq[i];
	        pq[i] = pq[j];
	        pq[j] = swap;
	        qp[pq[i]] = i;
	        qp[pq[j]] = j;
	    }

	    private void swim(int k) {
	        while (k > 1 && less(k/2, k)) {
	            exch(k, k/2);
	            k = k/2;
	        }
	    }

	    private void sink(int k) {
	        while (2*k <= n) {
	            int j = 2*k;
	            if (j < n && less(j, j+1)) j++;
	            if (!less(k, j)) break;
	            exch(k, j);
	            k = j;
	        }
	    }


	    /**
	     * Returns an iterator that iterates over the keys on the
	     * priority queue in descending order.
	     * The iterator doesn't implement {@code remove()} since it's optional.
	     *
	     * @return an iterator that iterates over the keys in descending order
	     */
	    public Iterator<Integer> iterator() {
	        return new HeapIterator();
	    }

	    private class HeapIterator implements Iterator<Integer> {
	        // create a new pq
	        private IndexMaxPQ<Key> copy;

	        // add all elements to copy of heap
	        // takes linear time since already in heap order so no keys move
	        public HeapIterator() {
	            copy = new IndexMaxPQ<Key>(pq.length - 1);
	            for (int i = 1; i <= n; i++)
	                copy.insert(pq[i], keys[pq[i]]);
	        }

	        public boolean hasNext()  { return !copy.isEmpty();                     }
	        public void remove()      { throw new UnsupportedOperationException();  }

	        public Integer next() {
	            if (!hasNext()) throw new NoSuchElementException();
	            return copy.delMax();
	        }
	    }

	}
	
	public static class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
	    private int maxN;        // maximum number of elements on PQ
	    private int n;           // number of elements on PQ
	    private int[] pq;        // binary heap using 1-based indexing
	    private int[] qp;        // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
	    private Key[] keys;      // keys[i] = priority of i

	    public IndexMinPQ(int maxN) {
	        if (maxN < 0) throw new IllegalArgumentException();
	        this.maxN = maxN;
	        n = 0;
	        keys = (Key[]) new Comparable[maxN + 1];    // make this of length maxN??
	        pq   = new int[maxN + 1];
	        qp   = new int[maxN + 1];                   // make this of length maxN??
	        for (int i = 0; i <= maxN; i++)
	            qp[i] = -1;
	    }

	    public boolean isEmpty() {
	        return n == 0;
	    }

	    public boolean contains(int i) {
	        validateIndex(i);
	        return qp[i] != -1;
	    }

	    public int size() {
	        return n;
	    }

	    /**
	     * Associates key with index {@code i}.
	     *
	     * @param  i an index
	     * @param  key the key to associate with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if there already is an item associated
	     *         with index {@code i}
	     */
	    public void insert(int i, Key key) {
	        validateIndex(i);
	        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
	        n++;
	        qp[i] = n;
	        pq[n] = i;
	        keys[i] = key;
	        swim(n);
	    }

	    /**
	     * Returns an index associated with a minimum key.
	     *
	     * @return an index associated with a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int minIndex() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return pq[1];
	    }

	    /**
	     * Returns a minimum key.
	     *
	     * @return a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public Key minKey() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return keys[pq[1]];
	    }

	    /**
	     * Removes a minimum key and returns its associated index.
	     * @return an index associated with a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int delMin() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        int min = pq[1];
	        exch(1, n--);
	        sink(1);
	        assert min == pq[n+1];
	        qp[min] = -1;        // delete
	        keys[min] = null;    // to help with garHashSete collection
	        pq[n+1] = -1;        // not needed
	        return min;
	    }

	    /**
	     * Returns the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to return
	     * @return the key associated with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public Key keyOf(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        else return keys[i];
	    }

	    /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void changeKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	        sink(qp[i]);
	    }

	    /**
	     * Decrease the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to decrease
	     * @param  key decrease the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key >= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void decreaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) < 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key strictly greater than the key in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	    }

	    /**
	     * Increase the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to increase
	     * @param  key increase the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key <= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void increaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) > 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key strictly less than the key in the priority queue");
	        keys[i] = key;
	        sink(qp[i]);
	    }

	    /**
	     * Remove the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to remove
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void delete(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        int index = qp[i];
	        exch(index, n--);
	        swim(index);
	        sink(index);
	        keys[i] = null;
	        qp[i] = -1;
	    }

	    // throw an IllegalArgumentException if i is an invalid index
	    private void validateIndex(int i) {
	        if (i < 0) throw new IllegalArgumentException("index is negative: " + i);
	        if (i >= maxN) throw new IllegalArgumentException("index >= capacity: " + i);
	    }

	    private boolean greater(int i, int j) {
	        return keys[pq[i]].compareTo(keys[pq[j]]) > 0;
	    }

	    private void exch(int i, int j) {
	        int swap = pq[i];
	        pq[i] = pq[j];
	        pq[j] = swap;
	        qp[pq[i]] = i;
	        qp[pq[j]] = j;
	    }

	    private void swim(int k) {
	        while (k > 1 && greater(k/2, k)) {
	            exch(k, k/2);
	            k = k/2;
	        }
	    }

	    private void sink(int k) {
	        while (2*k <= n) {
	            int j = 2*k;
	            if (j < n && greater(j, j+1)) j++;
	            if (!greater(k, j)) break;
	            exch(k, j);
	            k = j;
	        }
	    }

	    public Iterator<Integer> iterator() { return new HeapIterator(); }

	    private class HeapIterator implements Iterator<Integer> {
	        // create a new pq
	        private IndexMinPQ<Key> copy;

	        // add all elements to copy of heap
	        // takes linear time since already in heap order so no keys move
	        public HeapIterator() {
	            copy = new IndexMinPQ<Key>(pq.length - 1);
	            for (int i = 1; i <= n; i++)
	                copy.insert(pq[i], keys[pq[i]]);
	        }

	        public boolean hasNext()  { return !copy.isEmpty();                     }
	        public void remove()      { throw new UnsupportedOperationException();  }

	        public Integer next() {
	            if (!hasNext()) throw new NoSuchElementException();
	            return copy.delMin();
	        }
	    }

	}
}

// NOTES: 
// ASCII VALUE OF 'A': 65
// ASCII VALUE OF 'a': 97
// Range of long: 9 * 10^18
// ASCII VALUE OF '0': 48