Attention: we lost all the test cases for this problem, so instead of solving the problem, we need you to generate test cases. We're going to give you the answer, and you need to print a test case that produces the given answer. The original problem is in the following paragraph.

People don't use cash as often as they used to. Having a credit card solves some of the hassles of cash, such as having to receive change when you can't form the exact amount of money needed to purchase an item. Typically cashiers will give you as few coins as possible in change, but they don't have to. For example, if your change is 30 cents, a cashier could give you a 5 cent piece and a 25 cent piece, or they could give you three 10 cent pieces, or ten 1 cent pieces, two 5 cent pieces, and one 10 cent piece. Altogether there are 18 different ways to make 30 cents using only 1 cent pieces, 5 cent pieces, 10 cent pieces, and 25 cent pieces. Two ways are considered different if they contain a different number of at least one type of coin. Given the denominations of the coins and an amount of change to be made, how many different ways are there to make change?

As we mentioned before, we lost all the test cases for this problem, so we're actually going to give you the number of ways, and want you to produce a test case for which the number of ways is the given number. There could be many ways to achieve this (we guarantee there's always at least one), so you can print any, as long as it meets the constraints described below.

Input

Input will consist of a single integer A (1 ≤ A ≤ 105), the desired number of ways.

Output

In the first line print integers N and M (1 ≤ N ≤ 106, 1 ≤ M ≤ 10), the amount of change to be made, and the number of denominations, respectively.

Then print M integers D1, D2, ..., DM (1 ≤ Di ≤ 106), the denominations of the coins. All denominations must be distinct: for any i ≠ j we must have Di ≠ Dj.

If there are multiple tests, print any of them. You can print denominations in atbitrary order.

Examples

Input

18


Output

30 4
1 5 10 25


Input

3


Output

20 2
5 2


Input

314


Output

183 4
6 5 2 139
//package baobab;
import java.io.*;
import java.util.*;

public class B {

    public static final boolean DEBUG_PRINTS = false;

    public static void main(String[] args) {
        Solver solver = new Solver();
    }

    static class Solver {
        IO io;

        public Solver() {
            this.io = new IO();
            try {
                int numWays = io.nextInt();
                solve(numWays);
            } catch (RuntimeException e) {
                if (!e.getMessage().equals("Clean exit")) {
                    throw e;
                }
            } finally {
                io.close();
            }
        }

        /****************************** START READING HERE ********************************/

        void solve(int numWays) {
            if (numWays == 1) {
                io.println("1 1\n1");
                return;
            }
            int amount = 2*numWays - 2;
            io.println(amount + " " + 2);
            io.println("1 2");
        }

        /************************** UTILITY CODE BELOW THIS LINE **************************/

        long MOD = (long)1e9 + 7;

        List<Integer>[] toGraph(IO io, int n) {
            List<Integer>[] g = new ArrayList[n+1];
            for (int i=1; i<=n; i++) g[i] = new ArrayList<>();
            for (int i=1; i<=n-1; i++) {
                int a = io.nextInt();
                int b = io.nextInt();
                g[a].add(b);
                g[b].add(a);
            }
            return g;
        }

        class Point {
            int y;
            int x;

            public Point(int y, int x) {
                this.y = y;
                this.x = x;
            }
        }

        class IDval implements Comparable<IDval> {
            int id;
            long val;

            public IDval(int id, long val) {
                this.val = val;
                this.id = id;
            }

            @Override
            public int compareTo(IDval o) {
                if (this.val < o.val) return -1;
                if (this.val > o.val) return 1;
                return this.id - o.id;
            }
        }

        long pow(long base, int exp) {
            if (exp == 0) return 1L;
            long x = pow(base, exp/2);
            long ans = x * x;
            if (exp % 2 != 0) ans *= base;
            return ans;
        }

        long gcd(long... v) {
            /** Chained calls to Euclidean algorithm. */
            if (v.length == 1) return v[0];
            long ans = gcd(v[1], v[0]);
            for (int i=2; i<v.length; i++) {
                ans = gcd(ans, v[i]);
            }
            return ans;
        }

        long gcd(long a, long b) {
            /** Euclidean algorithm. */
            if (b == 0) return a;
            return gcd(b, a%b);
        }

        int[] generatePrimesUpTo(int last) {
            /* Sieve of Eratosthenes. Practically O(n). Values of 0 indicate primes. */
            int[] div = new int[last+1];
            for (int x=2; x<=last; x++) {
                if (div[x] > 0) continue;
                for (int u=2*x; u<=last; u+=x) {
                    div[u] = x;
                }
            }
            return div;
        }

        long lcm(long a, long b) {
            /** Least common multiple */
            return a * b / gcd(a,b);
        }

        private class ElementCounter {
            private HashMap<Long, Integer> elements;

            public ElementCounter() {
                elements = new HashMap<>();
            }

            public void add(long element) {
                int count = 1;
                if (elements.containsKey(element)) count += elements.get(element);
                elements.put(element, count);
            }

            public void remove(long element) {
                int count = elements.get(element);
                count--;
                if (count == 0) elements.remove(element);
                else elements.put(element, count);
            }

            public int get(long element) {
                if (!elements.containsKey(element)) return 0;
                return elements.get(element);
            }

            public int size() {
                return elements.size();
            }
        }

        class StringCounter {
            HashMap<String, Long> elements;

            public StringCounter() {
                elements = new HashMap<>();
            }

            public void add(String identifier) {
                long count = 1;
                if (elements.containsKey(identifier)) count += elements.get(identifier);
                elements.put(identifier, count);
            }

            public void remove(String identifier) {
                long count = elements.get(identifier);
                count--;
                if (count == 0) elements.remove(identifier);
                else elements.put(identifier, count);
            }

            public long get(String identifier) {
                if (!elements.containsKey(identifier)) return 0;
                return elements.get(identifier);
            }

            public int size() {
                return elements.size();
            }
        }

        class DisjointSet {
            /** Union Find / Disjoint Set data structure. */
            int[] size;
            int[] parent;
            int componentCount;

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

            public void join(int a, int b) {
                /* Find roots */
                int rootA = parent[a];
                int rootB = parent[b];
                while (rootA != parent[rootA]) rootA = parent[rootA];
                while (rootB != parent[rootB]) rootB = parent[rootB];

                if (rootA == rootB) {
                    /* Already in the same set */
                    return;
                }

                /* Merge smaller set into larger set. */
                if (size[rootA] > size[rootB]) {
                    size[rootA] += size[rootB];
                    parent[rootB] = rootA;
                } else {
                    size[rootB] += size[rootA];
                    parent[rootA] = rootB;
                }
                componentCount--;
            }

        }

        class LCAFinder {

            /* O(n log n) Initialize: new LCAFinder(graph)
             * O(log n) Queries: find(a,b) returns lowest common ancestor for nodes a and b */

            int[] nodes;
            int[] depths;
            int[] entries;
            int pointer;
            FenwickMin fenwick;

            public LCAFinder(List<Integer>[] graph) {
                this.nodes = new int[(int)10e6];
                this.depths = new int[(int)10e6];
                this.entries = new int[graph.length];
                this.pointer = 1;
                boolean[] visited = new boolean[graph.length+1];
                dfs(1, 0, graph, visited);
                fenwick = new FenwickMin(pointer-1);
                for (int i=1; i<pointer; i++) {
                    fenwick.set(i, depths[i] * 1000000L + i);
                }
            }

            private void dfs(int node, int depth, List<Integer>[] graph, boolean[] visited) {
                visited[node] = true;
                entries[node] = pointer;
                nodes[pointer] = node;
                depths[pointer] = depth;
                pointer++;
                for (int neighbor : graph[node]) {
                    if (visited[neighbor]) continue;
                    dfs(neighbor, depth+1, graph, visited);
                    nodes[pointer] = node;
                    depths[pointer] = depth;
                    pointer++;
                }
            }

            public int find(int a, int b) {
                int left = entries[a];
                int right = entries[b];
                if (left > right) {
                    int temp = left;
                    left = right;
                    right = temp;
                }
                long mixedBag = fenwick.getMin(left, right);
                int index = (int) (mixedBag % 1000000L);
                return nodes[index];
            }
        }

        class FenwickMin {
            long n;
            long[] original;
            long[] bottomUp;
            long[] topDown;

            public FenwickMin(int n) {
                this.n = n;
                original = new long[n+2];
                bottomUp = new long[n+2];
                topDown = new long[n+2];
            }

            public void set(int modifiedNode, long value) {
                long replaced = original[modifiedNode];
                original[modifiedNode] = value;
                // Update left tree
                int i = modifiedNode;
                long v = value;
                while (i <= n) {
                    if (v > bottomUp[i]) {
                        if (replaced == bottomUp[i]) {
                            v = Math.min(v, original[i]);
                            for (int r=1 ;; r++) {
                                int x = (i&-i)>>>r;
                                if (x == 0) break;
                                int child = i-x;
                                v = Math.min(v, bottomUp[child]);
                            }
                        } else break;
                    }
                    if (v == bottomUp[i]) break;
                    bottomUp[i] = v;
                    i += (i&-i);
                }
                // Update right tree
                i = modifiedNode;
                v = value;
                while (i > 0) {
                    if (v > topDown[i]) {
                        if (replaced == topDown[i]) {
                            v = Math.min(v, original[i]);
                            for (int r=1 ;; r++) {
                                int x = (i&-i)>>>r;
                                if (x == 0) break;
                                int child = i+x;
                                if (child > n+1) break;
                                v = Math.min(v, topDown[child]);
                            }
                        } else break;
                    }
                    if (v == topDown[i]) break;
                    topDown[i] = v;
                    i -= (i&-i);
                }
            }

            public long getMin(int a, int b) {
                long min = original[a];
                int prev = a;
                int curr = prev + (prev&-prev); // parent right hand side
                while (curr <= b) {
                    min = Math.min(min, topDown[prev]); // value from the other tree
                    prev = curr;
                    curr = prev + (prev&-prev);;
                }
                min = Math.min(min, original[prev]);
                prev = b;
                curr = prev - (prev&-prev); // parent left hand side
                while (curr >= a) {
                    min = Math.min(min,bottomUp[prev]); // value from the other tree
                    prev = curr;
                    curr = prev - (prev&-prev);
                }
                return min;
            }

        }

        class FenwickSum {
            public long[] d;

            public FenwickSum(int n) {
                d=new long[n+1];
            }

            /** a[0] must be unused. */
            public FenwickSum(long[] a) {
                d=new long[a.length];
                for (int i=1; i<a.length; i++) {
                    modify(i, a[i]);
                }
            }

            /** Do not modify i=0. */
            void modify(int i, long v) {
                while (i<d.length) {
                    d[i] += v;
                    // Move to next uplink on the RIGHT side of i
                    i += (i&-i);
                }
            }

            /** Returns sum from a to b, *BOTH* inclusive. */
            long getSum(int a, int b) {
                return getSum(b) - getSum(a-1);
            }

            private long getSum(int i) {
                long sum = 0;
                while (i>0) {
                    sum += d[i];
                    // Move to next uplink on the LEFT side of i
                    i -= (i&-i);
                }
                return sum;
            }
        }

        class SegmentTree {
            /** Query sums with log(n) modifyRange */
            int N;
            long[] p;

            public SegmentTree(int n) {
                /* TODO: Test that this works. */
                for (N=2; N<n; N++) N *= 2;
                p = new long[2*N];
            }

            public void modifyRange(int a, int b, long change) {
                muuta(a, change);
                muuta(b+1, -change);
            }

            void muuta(int k, long muutos) {
                k += N;
                p[k] += muutos;
                for (k /= 2; k >= 1; k /= 2) {
                    p[k] = p[2*k] + p[2*k+1];
                }
            }

            public long get(int k) {
                int a = N;
                int b = k+N;
                long s = 0;
                while (a <= b) {
                    if (a%2 == 1) s += p[a++];
                    if (b%2 == 0) s += p[b--];
                    a /= 2;
                    b /= 2;
                }
                return s;
            }

        }

        class Zalgo {

            public int pisinEsiintyma(String haku, String kohde) {
                char[] s = new char[haku.length() + 1 + kohde.length()];
                for (int i=0; i<haku.length(); i++) {
                    s[i] = haku.charAt(i);
                }
                int j = haku.length();
                s[j++] = '#';
                for (int i=0; i<kohde.length(); i++) {
                    s[j++] = kohde.charAt(i);
                }
                int[] z = toZarray(s);
                int max = 0;
                for (int i=haku.length(); i<z.length; i++) {
                    max = Math.max(max, z[i]);
                }
                return max;
            }

            public int[] toZarray(char[] s) {
                int n = s.length;
                int[] z = new int[n];
                int a = 0, b = 0;
                for (int i = 1; i < n; i++) {
                    if (i > b) {
                        for (int j = i; j < n && s[j - i] == s[j]; j++)  z[i]++;
                    }
                    else {
                        z[i] = z[i - a];
                        if (i + z[i - a] > b) {
                            for (int j = b + 1; j < n && s[j - i] == s[j]; j++) z[i]++;
                            a = i;
                            b = i + z[i] - 1;
                        }
                    }
                }
                return z;
            }

            public List<Integer> getStartIndexesWhereWordIsFound(String haku, String kohde) {
                // this is alternative use case
                char[] s = new char[haku.length() + 1 + kohde.length()];
                for (int i=0; i<haku.length(); i++) {
                    s[i] = haku.charAt(i);
                }
                int j = haku.length();
                s[j++] = '#';
                for (int i=0; i<kohde.length(); i++) {
                    s[j++] = kohde.charAt(i);
                }
                int[] z = toZarray(s);
                List<Integer> indexes = new ArrayList<>();
                for (int i=haku.length(); i<z.length; i++) {
                    if (z[i] < haku.length()) continue;
                    indexes.add(i);
                }
                return indexes;
            }

        }

        class StringHasher {

            class HashedString {
                long[] hashes;
                long[] modifiers;

                public HashedString(long[] hashes, long[] modifiers) {
                    this.hashes = hashes;
                    this.modifiers = modifiers;
                }
            }

            long P;
            long M;

            public StringHasher() {
                initializePandM();
            }

            HashedString hashString(String s) {
                int n = s.length();
                long[] hashes = new long[n];
                long[] modifiers = new long[n];
                hashes[0] = s.charAt(0);
                modifiers[0] = 1;
                for (int i=1; i<n; i++) {
                    hashes[i] = (hashes[i-1] * P + s.charAt(i)) % M;
                    modifiers[i] = (modifiers[i-1] * P) % M;
                }
                return new HashedString(hashes, modifiers);
            }

            /**
             * Indices are inclusive.
             */
            long getHash(HashedString hashedString, int startIndex, int endIndex) {
                long[] hashes = hashedString.hashes;
                long[] modifiers = hashedString.modifiers;
                long result = hashes[endIndex];
                if (startIndex > 0) result -= (hashes[startIndex-1] * modifiers[endIndex-startIndex+1]) % M;
                if (result < 0) result += M;
                return result;
            }

            // Less interesting methods below



            /**
             * Efficient for 2 input parameter strings in particular.
             */
            HashedString[] hashString(String first, String second) {
                HashedString[] array = new HashedString[2];
                int n = first.length();
                long[] modifiers = new long[n];
                modifiers[0] = 1;

                long[] firstHashes = new long[n];
                firstHashes[0] = first.charAt(0);
                array[0] = new HashedString(firstHashes, modifiers);

                long[] secondHashes = new long[n];
                secondHashes[0] = second.charAt(0);
                array[1] = new HashedString(secondHashes, modifiers);

                for (int i=1; i<n; i++) {
                    modifiers[i] = (modifiers[i-1] * P) % M;
                    firstHashes[i] = (firstHashes[i-1] * P + first.charAt(i)) % M;
                    secondHashes[i] = (secondHashes[i-1] * P + second.charAt(i)) % M;
                }
                return array;
            }

            /**
             * Efficient for 3+ strings
             * More efficient than multiple hashString calls IF strings are same length.
             */
            HashedString[] hashString(String... strings) {
                HashedString[] array = new HashedString[strings.length];
                int n = strings[0].length();
                long[] modifiers = new long[n];
                modifiers[0] = 1;
                for (int j=0; j<strings.length; j++) {
                    // if all strings are not same length, defer work to another method
                    if (strings[j].length() != n) {
                        for (int i=0; i<n; i++) {
                            array[i] = hashString(strings[i]);
                        }
                        return array;
                    }

                    // otherwise initialize stuff
                    long[] hashes = new long[n];
                    hashes[0] = strings[j].charAt(0);
                    array[j] = new HashedString(hashes, modifiers);
                }
                for (int i=1; i<n; i++) {
                    modifiers[i] = (modifiers[i-1] * P) % M;
                    for (int j=0; j<strings.length; j++) {
                        String s = strings[j];
                        long[] hashes = array[j].hashes;
                        hashes[i] = (hashes[i-1] * P + s.charAt(i)) % M;
                    }
                }
                return array;
            }

            void initializePandM() {
                ArrayList<Long> modOptions = new ArrayList<>(20);
                modOptions.add(353873237L);
                modOptions.add(353875897L);
                modOptions.add(353878703L);
                modOptions.add(353882671L);
                modOptions.add(353885303L);
                modOptions.add(353888377L);
                modOptions.add(353893457L);
                P = modOptions.get(new Random().nextInt(modOptions.size()));

                modOptions.clear();
                modOptions.add(452940277L);
                modOptions.add(452947687L);
                modOptions.add(464478431L);
                modOptions.add(468098221L);
                modOptions.add(470374601L);
                modOptions.add(472879717L);
                modOptions.add(472881973L);
                M = modOptions.get(new Random().nextInt(modOptions.size()));

            }
        }

        private static class Prob {

            /** For heavy calculations on probabilities, this class
             *  provides more accuracy & efficiency than doubles.
             *  Math explained: https://en.wikipedia.org/wiki/Log_probability
             *  Quick start:
             *      - Instantiate probabilities, eg. Prob a = new Prob(0.75)
             *      - add(), multiply() return new objects, can perform on nulls & NaNs.
             *      - get() returns probability as a readable double */

            /** Logarithmized probability. Note: 0% represented by logP NaN. */
            private double logP;

            /** Construct instance with real probability. */
            public Prob(double real) {
                if (real > 0) this.logP = Math.log(real);
                else this.logP = Double.NaN;
            }

            /** Construct instance with already logarithmized value. */
            static boolean dontLogAgain = true;
            public Prob(double logP, boolean anyBooleanHereToChooseThisConstructor) {
                this.logP = logP;
            }

            /** Returns real probability as a double. */
            public double get() {
                return Math.exp(logP);
            }

            @Override
            public String toString() {
                return ""+get();
            }

            /***************** STATIC METHODS BELOW ********************/

            /** Note: returns NaN only when a && b are both NaN/null. */
            public static Prob add(Prob a, Prob b) {
                if (nullOrNaN(a) && nullOrNaN(b)) return new Prob(Double.NaN, dontLogAgain);
                if (nullOrNaN(a)) return copy(b);
                if (nullOrNaN(b)) return copy(a);

                double x = Math.max(a.logP, b.logP);
                double y = Math.min(a.logP, b.logP);
                double sum = x + Math.log(1 + Math.exp(y - x));
                return new Prob(sum, dontLogAgain);
            }

            /** Note: multiplying by null or NaN produces NaN (repping 0% real prob). */
            public static Prob multiply(Prob a, Prob b) {
                if (nullOrNaN(a) || nullOrNaN(b)) return new Prob(Double.NaN, dontLogAgain);
                return new Prob(a.logP + b.logP, dontLogAgain);
            }

            /** Returns true if p is null or NaN. */
            private static boolean nullOrNaN(Prob p) {
                return (p == null || Double.isNaN(p.logP));
            }

            /** Returns a new instance with the same value as original. */
            private static Prob copy(Prob original) {
                return new Prob(original.logP, dontLogAgain);
            }
        }

        public class StronglyConnectedComponents {

            /** Kosaraju's algorithm */

            ArrayList<Integer>[] forw;
            ArrayList<Integer>[] bacw;

            /** Use: getCount(2, new int[] {1,2}, new int[] {2,1}) */
            public int getCount(int n, int[] mista, int[] minne) {
                forw = new ArrayList[n+1];
                bacw = new ArrayList[n+1];
                for (int i=1; i<=n; i++) {
                    forw[i] = new ArrayList<Integer>();
                    bacw[i] = new ArrayList<Integer>();
                }
                for (int i=0; i<mista.length; i++) {
                    int a = mista[i];
                    int b = minne[i];
                    forw[a].add(b);
                    bacw[b].add(a);
                }
                int count = 0;
                List<Integer> list = new ArrayList<Integer>();
                boolean[] visited = new boolean[n+1];
                for (int i=1; i<=n; i++) {
                    dfsForward(i, visited, list);
                }
                visited = new boolean[n+1];
                for (int i=n-1; i>=0; i--) {
                    int node = list.get(i);
                    if (visited[node]) continue;
                    count++;
                    dfsBackward(node, visited);
                }
                return count;
            }

            public void dfsForward(int i, boolean[] visited, List<Integer> list) {
                if (visited[i]) return;
                visited[i] = true;
                for (int neighbor : forw[i]) {
                    dfsForward(neighbor, visited, list);
                }
                list.add(i);
            }

            public void dfsBackward(int i, boolean[] visited) {
                if (visited[i]) return;
                visited[i] = true;
                for (int neighbor : bacw[i]) {
                    dfsBackward(neighbor, visited);
                }
            }
        }

        class DrawGrid {

            void draw(boolean[][] d) {
                System.out.print("  ");
                for (int x=0; x<d[0].length; x++) {
                    System.out.print(" " + x + " ");
                }
                System.out.println("");
                for (int y=0; y<d.length; y++) {
                    System.out.print(y + " ");
                    for (int x=0; x<d[0].length; x++) {
                        System.out.print((d[y][x] ? "[x]" : "[ ]"));
                    }
                    System.out.println("");
                }
            }

            void draw(int[][] d) {
                int max = 1;
                for (int y=0; y<d.length; y++) {
                    for (int x=0; x<d[0].length; x++) {
                        max = Math.max(max, ("" + d[y][x]).length());
                    }
                }
                System.out.print("  ");
                String format = "%" + (max+2) + "s";
                for (int x=0; x<d[0].length; x++) {
                    System.out.print(String.format(format, x) + " ");
                }
                format = "%" + (max) + "s";
                System.out.println("");
                for (int y=0; y<d.length; y++) {
                    System.out.print(y + " ");
                    for (int x=0; x<d[0].length; x++) {
                        System.out.print(" [" + String.format(format, (d[y][x])) + "]");
                    }
                    System.out.println("");
                }
            }

        }

        class BaseConverter {

            /* Palauttaa luvun esityksen kannassa base */
            public String convert(Long number, int base) {
                return Long.toString(number, base);
            }

            /* Palauttaa luvun esityksen kannassa baseTo, kun annetaan luku Stringinä kannassa baseFrom */
            public String convert(String number, int baseFrom, int baseTo) {
                return Long.toString(Long.parseLong(number, baseFrom), baseTo);
            }

            /* Tulkitsee kannassa base esitetyn luvun longiksi (kannassa 10) */
            public long longify(String number, int baseFrom) {
                return Long.parseLong(number, baseFrom);
            }
        }

        class Binary implements Comparable<Binary> {

            /**
             * Use example: Binary b = new Binary(Long.toBinaryString(53249834L));
             *
             * When manipulating small binary strings, instantiate new Binary(string)
             * When just reading large binary strings, instantiate new Binary(string,true)
             * get(int i) returns a character '1' or '0', not an int.
             */

            private boolean[] d;
            private int first; // Starting from left, the first (most remarkable) '1'
            public int length;


            public Binary(String binaryString) {
                this(binaryString, false);
            }
            public Binary(String binaryString, boolean initWithMinArraySize) {
                length = binaryString.length();
                int size = Math.max(2*length, 1);
                first = length/4;
                if (initWithMinArraySize) {
                    first = 0;
                    size = Math.max(length, 1);
                }
                d = new boolean[size];
                for (int i=0; i<length; i++) {
                    if (binaryString.charAt(i) == '1') d[i+first] = true;
                }
            }

            public void addFirst(char c) {
                if (first-1 < 0) doubleArraySize();
                first--;
                d[first] = (c == '1' ? true : false);
                length++;
            }

            public void addLast(char c) {
                if (first+length >= d.length) doubleArraySize();
                d[first+length] = (c == '1' ? true : false);
                length++;
            }

            private void doubleArraySize() {
                boolean[] bigArray = new boolean[(d.length+1) * 2];
                int newFirst = bigArray.length / 4;
                for (int i=0; i<length; i++) {
                    bigArray[i + newFirst] = d[i + first];
                }
                first = newFirst;
                d = bigArray;
            }

            public boolean flip(int i) {
                boolean value = (this.d[first+i] ? false : true);
                this.d[first+i] = value;
                return value;
            }

            public void set(int i, char c) {
                boolean value = (c == '1' ? true : false);
                this.d[first+i] = value;
            }

            public char get(int i) {
                return (this.d[first+i] ? '1' : '0');
            }

            @Override
            public int compareTo(Binary o) {
                if (this.length != o.length) return this.length - o.length;
                int len = this.length;
                for (int i=0; i<len; i++) {
                    int diff = this.get(i) - o.get(i);
                    if (diff != 0) return diff;
                }
                return 0;
            }

            @Override
            public String toString() {
                StringBuilder sb = new StringBuilder();
                for (int i=0; i<length; i++) {
                    sb.append(d[i+first] ? '1' : '0');
                }
                return sb.toString();
            }


        }

        class BinomialCoefficients {
            /** Total number of K sized unique combinations from pool of size N (unordered)
             N! / ( K! (N - K)! )   */

            /** For simple queries where output fits in long. */
            public long biCo(long n, long k) {
                long r = 1;
                if (k > n) return 0;
                for (long d = 1; d <= k; d++) {
                    r *= n--;
                    r /= d;
                }
                return r;
            }

            /** For multiple queries with same n, different k. */
            public long[] precalcBinomialCoefficientsK(int n, int maxK) {
                long v[] = new long[maxK+1];
                v[0] = 1; // nC0 == 1
                for (int i=1; i<=n; i++) {
                    for (int j=Math.min(i,maxK); j>0; j--) {
                        v[j] = v[j] + v[j-1]; // Pascal's triangle
                    }
                }
                return v;
            }

            /** When output needs % MOD. */
            public long[] precalcBinomialCoefficientsK(int n, int k, long M) {
                long v[] = new long[k+1];
                v[0] = 1; // nC0 == 1
                for (int i=1; i<=n; i++) {
                    for (int j=Math.min(i,k); j>0; j--) {
                        v[j] = v[j] + v[j-1]; // Pascal's triangle
                        v[j] %= M;
                    }
                }
                return v;
            }
        }

        class Trie {
            int N;
            int Z;
            int nextFreeId;
            int[][] pointers;
            boolean[] end;

            /** maxLenSum = maximum possible sum of length of words */
            public Trie(int maxLenSum, int alphabetSize) {
                this.N = maxLenSum;
                this.Z = alphabetSize;
                this.nextFreeId = 1;
                pointers = new int[N+1][alphabetSize];
                end = new boolean[N+1];
            }

            public void addWord(String word) {
                int curr = 0;
                for (int j=0; j<word.length(); j++) {
                    int c = word.charAt(j) - 'a';
                    int next = pointers[curr][c];
                    if (next == 0) {
                        next = nextFreeId++;
                        pointers[curr][c] = next;
                    }
                    curr = next;
                }
                int last = word.charAt(word.length()-1) - 'a';
                end[last] = true;
            }

            public boolean hasWord(String word) {
                int curr = 0;
                for (int j=0; j<word.length(); j++) {
                    int c = word.charAt(j) - 'a';
                    int next = pointers[curr][c];
                    if (next == 0) return false;
                    curr = next;
                }
                int last = word.charAt(word.length()-1) - 'a';
                return end[last];
            }

        }

        private class IO extends PrintWriter {
            private InputStreamReader r;
            private static final int BUFSIZE = 1 << 15;
            private char[] buf;
            private int bufc;
            private int bufi;
            private StringBuilder sb;

            public IO() {
                super(new BufferedOutputStream(System.out));
                r = new InputStreamReader(System.in);

                buf = new char[BUFSIZE];
                bufc = 0;
                bufi = 0;
                sb = new StringBuilder();
            }

            /** Print, flush, return nextInt. */
            private int queryInt(String s) {
                io.println(s);
                io.flush();
                return nextInt();
            }

            /** Print, flush, return nextLong. */
            private long queryLong(String s) {
                io.println(s);
                io.flush();
                return nextLong();
            }

            /** Print, flush, return next word. */
            private String queryNext(String s) {
                io.println(s);
                io.flush();
                return next();
            }

            private void fillBuf() throws IOException {
                bufi = 0;
                bufc = 0;
                while(bufc == 0) {
                    bufc = r.read(buf, 0, BUFSIZE);
                    if(bufc == -1) {
                        bufc = 0;
                        return;
                    }
                }
            }

            private boolean pumpBuf() throws IOException {
                if(bufi == bufc) {
                    fillBuf();
                }
                return bufc != 0;
            }

            private boolean isDelimiter(char c) {
                return c == ' ' || c == '\t' || c == '\n' || c == '\r' || c == '\f';
            }

            private void eatDelimiters() throws IOException {
                while(true) {
                    if(bufi == bufc) {
                        fillBuf();
                        if(bufc == 0) throw new RuntimeException("IO: Out of input.");
                    }

                    if(!isDelimiter(buf[bufi])) break;
                    ++bufi;
                }
            }

            public String next() {
                try {
                    sb.setLength(0);

                    eatDelimiters();
                    int start = bufi;

                    while(true) {
                        if(bufi == bufc) {
                            sb.append(buf, start, bufi - start);
                            fillBuf();
                            start = 0;
                            if(bufc == 0) break;
                        }

                        if(isDelimiter(buf[bufi])) break;
                        ++bufi;
                    }

                    sb.append(buf, start, bufi - start);

                    return sb.toString();
                } catch(IOException e) {
                    throw new RuntimeException("IO.next: Caught IOException.");
                }
            }

            public int nextInt() {
                try {
                    int ret = 0;

                    eatDelimiters();

                    boolean positive = true;
                    if(buf[bufi] == '-') {
                        ++bufi;
                        if(!pumpBuf()) throw new RuntimeException("IO.nextInt: Invalid int.");
                        positive = false;
                    }

                    boolean first = true;
                    while(true) {
                        if(!pumpBuf()) break;
                        if(isDelimiter(buf[bufi])) {
                            if(first) throw new RuntimeException("IO.nextInt: Invalid int.");
                            break;
                        }
                        first = false;

                        if(buf[bufi] >= '0' && buf[bufi] <= '9') {
                            if(ret < -214748364) throw new RuntimeException("IO.nextInt: Invalid int.");
                            ret *= 10;
                            ret -= (int)(buf[bufi] - '0');
                            if(ret > 0) throw new RuntimeException("IO.nextInt: Invalid int.");
                        } else {
                            throw new RuntimeException("IO.nextInt: Invalid int.");
                        }

                        ++bufi;
                    }

                    if(positive) {
                        if(ret == -2147483648) throw new RuntimeException("IO.nextInt: Invalid int.");
                        ret = -ret;
                    }

                    return ret;
                } catch(IOException e) {
                    throw new RuntimeException("IO.nextInt: Caught IOException.");
                }
            }

            public long nextLong() {
                try {
                    long ret = 0;

                    eatDelimiters();

                    boolean positive = true;
                    if(buf[bufi] == '-') {
                        ++bufi;
                        if(!pumpBuf()) throw new RuntimeException("IO.nextLong: Invalid long.");
                        positive = false;
                    }

                    boolean first = true;
                    while(true) {
                        if(!pumpBuf()) break;
                        if(isDelimiter(buf[bufi])) {
                            if(first) throw new RuntimeException("IO.nextLong: Invalid long.");
                            break;
                        }
                        first = false;

                        if(buf[bufi] >= '0' && buf[bufi] <= '9') {
                            if(ret < -922337203685477580L) throw new RuntimeException("IO.nextLong: Invalid long.");
                            ret *= 10;
                            ret -= (long)(buf[bufi] - '0');
                            if(ret > 0) throw new RuntimeException("IO.nextLong: Invalid long.");
                        } else {
                            throw new RuntimeException("IO.nextLong: Invalid long.");
                        }

                        ++bufi;
                    }

                    if(positive) {
                        if(ret == -9223372036854775808L) throw new RuntimeException("IO.nextLong: Invalid long.");
                        ret = -ret;
                    }

                    return ret;
                } catch(IOException e) {
                    throw new RuntimeException("IO.nextLong: Caught IOException.");
                }
            }

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

        }

        void print(Object output) {
            io.println(output);
        }

        void done(Object output) {
            print(output);
            done();
        }

        void done() {
            io.close();
            throw new RuntimeException("Clean exit");
        }

        long min(long... v) {
            long ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        double min(double... v) {
            double ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        int min(int... v) {
            int ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        long max(long... v) {
            long ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        double max(double... v) {
            double ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        int max(int... v) {
            int ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        void debug(Object output) {
            if (DEBUG_PRINTS) {
                System.out.println(output);
            }
        }
    }

}



