using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using Library;

namespace Program
{
    public static class ProblemA
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var startTime = DateTime.Now;

            var N = (int)NN;
            var D = (int)NN;
            var H = (int)NN;
            var S = NSList(N);
            var M = (int)NN;
            var yxd = Repeat(0, M).Select(_ => new { y = (int)NN, x = (int)NN, d = (int)NN }).ToArray();

            var fieldItem = new int[N, N];
            const int JEWEL = 10000;
            const int WALL = 20000;
            const int ENEMY = 30000;
            const int FIRE = 40000;
            var jewelNum = 0;
            for (var i = 0; i < N; ++i)
            {
                for (var j = 0; j < N; ++j)
                {
                    switch (S[i][j])
                    {
                        case '#': fieldItem[i, j] = WALL; break;
                        case 'J': fieldItem[i, j] = ++jewelNum; break;
                        case 'E': fieldItem[i, j] = ENEMY; break;
                        case 'F': fieldItem[i, j] = FIRE; break;
                    }
                }
            }
            var startX = 0;
            var startY = 0;
            var keyX = 0;
            var keyY = 0;
            var goalX = 0;
            var goalY = 0;
            for (var i = 0; i < N; ++i)
            {
                for (var j = 0; j < N; ++j)
                {
                    if (S[i][j] == 'S')
                    {
                        startY = i;
                        startX = j;
                    }
                    if (S[i][j] == 'G')
                    {
                        goalY = i;
                        goalX = j;
                    }
                    if (S[i][j] == 'K')
                    {
                        keyY = i;
                        keyX = j;
                    }
                }
            }
            var diff_start_key = Abs(startY - keyY) + Abs(startX - keyX);
            var diff_start_goal = Abs(startY - goalY) + Abs(startX - goalX);
            var diff_key_goal = Abs(keyY - goalY) + Abs(keyX - goalX);

            var effect = new List<int>[N, N];
            for (var i = 0; i < N; ++i) for (var j = 0; j < N; ++j) effect[i, j] = new List<int>();
            var tantiki = new int[N, N];
            tantiki.Fill(-1);
            for (var i = 0; i < M; ++i) tantiki[yxd[i].y, yxd[i].x] = i;
            for (var i = 0; i < N; ++i)
            {
                var before = -1;
                for (var j = 0; j < N; ++j)
                {
                    if (before != -1) effect[i, j].Add(before);
                    if (tantiki[i, j] != -1) before = tantiki[i, j];
                    if (fieldItem[i, j] == WALL) before = -1;
                }
                before = -1;
                for (var j = N - 1; j >= 0; --j)
                {
                    if (before != -1) effect[i, j].Add(before);
                    if (tantiki[i, j] != -1) before = tantiki[i, j];
                    if (fieldItem[i, j] == WALL) before = -1;
                }
            }
            for (var j = 0; j < N; ++j)
            {
                var before = -1;
                for (var i = 0; i < N; ++i)
                {
                    if (before != -1) effect[i, j].Add(before);
                    if (tantiki[i, j] != -1) before = tantiki[i, j];
                    if (fieldItem[i, j] == WALL) before = -1;
                }
                before = -1;
                for (var i = N - 1; i >= 0; --i)
                {
                    if (before != -1) effect[i, j].Add(before);
                    if (tantiki[i, j] != -1) before = tantiki[i, j];
                    if (fieldItem[i, j] == WALL) before = -1;
                }
            }
            if (diff_start_key > diff_key_goal)
            {
                var kariLen = 0;
                {
                    var dp = new int[N, N];
                    var ndp = new int[N, N];
                    var ansField = new List<string[,]>();
                    dp[keyY, keyX] = H;
                    var finishTurn = 0;
                    var maxH = 0;
                    for (var turn = 1; turn < 300; ++turn)
                    {
                        ndp.Fill(0);
                        var ans = new string[N, N];
                        for (var i = 0; i < N; ++i)
                        {
                            for (var j = 0; j < N; ++j)
                            {
                                if (fieldItem[i, j] == WALL || fieldItem[i, j] == ENEMY) continue;
                                var nh = dp[i, j];
                                var ope = "S";
                                if (i > 0 && nh.Chmax(dp[i - 1, j])) ope = "M D";
                                if (i < N - 1 && nh.Chmax(dp[i + 1, j])) ope = "M U";
                                if (j > 0 && nh.Chmax(dp[i, j - 1])) ope = "M R";
                                if (j < N - 1 && nh.Chmax(dp[i, j + 1])) ope = "M L";
                                --nh;
                                foreach (var item in effect[i, j])
                                {
                                    if (turn % yxd[item].d == 0)
                                    {
                                        nh -= D;
                                    }
                                }
                                ndp[i, j] = nh;
                                ans[i, j] = ope;
                            }
                        }
                        var t = dp; dp = ndp; ndp = t;
                        if (maxH.Chmax(dp[goalY, goalX])) finishTurn = turn;
                        ansField.Add(ans);
                    }
                    kariLen = (int)((H - maxH) * 1.1);
                }
                var pool1 = new List<LIB_Bitset>(10000);
                var pool2 = new List<LIB_Bitset>(10000);

                Func<int, int, double, bool, (int lastHealth, int count, string[], int turn)> calc = (startTurn, startHelath, gain, dry) =>
                {
                    var dp = new (double health, int count, LIB_Bitset got)[N, N];
                    var ndp = new (double health, int count, LIB_Bitset got)[N, N];
                    var ansField = new List<string[,]>();
                    dp[startY, startX] = (startHelath, 0, new LIB_Bitset(jewelNum + 1));
                    var maxH = 0.0;
                    var maxCount = 0;
                    var finishTurn = 0;
                    for (var turn = startTurn; turn < 1000; ++turn)
                    {
                        ndp.Fill((0, 0, null));
                        string[,] ans = null;
                        if (!dry) ans = new string[N, N];
                        for (var i = 0; i < N; ++i)
                        {
                            for (var j = 0; j < N; ++j)
                            {
                                if (fieldItem[i, j] == WALL || fieldItem[i, j] == ENEMY) continue;
                                var nh = dp[i, j].health;
                                var ope = "S";
                                var count = dp[i, j].count;
                                var got = dp[i, j].got;
                                if (!(got is null)) pool2.Add(got);
                                if (i > 0 && nh.Chmax(dp[i - 1, j].health))
                                {
                                    ope = "M D";
                                    count = dp[i - 1, j].count;
                                    got = dp[i - 1, j].got;
                                }
                                if (i < N - 1 && nh.Chmax(dp[i + 1, j].health))
                                {
                                    ope = "M U";
                                    count = dp[i + 1, j].count;
                                    got = dp[i + 1, j].got;
                                }
                                if (j > 0 && nh.Chmax(dp[i, j - 1].health))
                                {
                                    ope = "M R";
                                    count = dp[i, j - 1].count;
                                    got = dp[i, j - 1].got;
                                }
                                if (j < N - 1 && nh.Chmax(dp[i, j + 1].health))
                                {
                                    ope = "M L";
                                    count = dp[i, j + 1].count;
                                    got = dp[i, j + 1].got;
                                }
                                nh -= 1;
                                foreach (var item in effect[i, j])
                                {
                                    if (turn % yxd[item].d == 0) nh -= D;
                                }
                                if (nh <= 0) continue;
                                if (pool1.Count > 0)
                                {
                                    var nextgot = pool1[pool1.Count - 1];
                                    pool1.RemoveAt(pool1.Count - 1);
                                    for (var k = 0; k < nextgot.ary.Length; ++k)
                                    {
                                        nextgot.ary[k] = got.ary[k];
                                    }
                                    got = nextgot;
                                }
                                else
                                {
                                    got = got | 0;
                                }
                                if (0 < fieldItem[i, j] && fieldItem[i, j] <= JEWEL && !got[fieldItem[i, j]])
                                {
                                    nh += gain;
                                    ++count;
                                    got[fieldItem[i, j]] = true;
                                }
                                ndp[i, j] = (nh, count, got);
                                if (!dry) ans[i, j] = ope;
                            }
                        }
                        var t = dp; dp = ndp; ndp = t;
                        var t3 = pool1; pool1 = pool2; pool2 = t3;

                        if (maxH.Chmax(dp[keyY, keyX].health))
                        {
                            maxCount = dp[keyY, keyX].count;
                            finishTurn = ansField.Count;
                        }
                        ansField.Add(ans);
                    }
                    var firstAnswer = new List<string>();
                    if (!dry)
                    {
                        var nowx = keyX;
                        var nowy = keyY;
                        for (var turn = finishTurn; turn >= 0; --turn)
                        {
                            var ope = ansField[turn][nowy, nowx];
                            firstAnswer.Add(ope);
                            if (ope == "M D") --nowy;
                            else if (ope == "M U") ++nowy;
                            else if (ope == "M R") --nowx;
                            else if (ope == "M L") ++nowx;
                        }
                        firstAnswer.Reverse();
                    }
                    return ((int)maxH, maxCount, firstAnswer.ToArray(), finishTurn + 1);
                };
                {
                    var left = 8.0;
                    var right = 12.0;
                    var gain = 0.0;
                    var minokgain = double.MaxValue;
                    while (true)
                    {
                        var turnTime = DateTime.Now;
                        var mid = (xorshift % 2000) / 1000.0 + 9.0;
                        var res = calc(1, H, mid, true);
                        var hp = res.lastHealth - res.count * mid;
                        if (hp >= kariLen)
                        {
                            if (minokgain.Chmin(hp)) gain = mid;
                        }
                        var elapsed = (DateTime.Now - turnTime).TotalMilliseconds;
                        if ((DateTime.Now - startTime).TotalMilliseconds + elapsed * 2 > 2800) break;
                    }
                    var ans = calc(1, H, gain, false);

                    var dp = new int[N, N];
                    var ndp = new int[N, N];
                    var ansField = new List<string[,]>();
                    dp[keyY, keyX] = (int)(ans.lastHealth - ans.count * gain);
                    var finishTurn = 0;
                    var maxH = 0;
                    for (var turn = ans.turn + 1; turn < 1000; ++turn)
                    {
                        ndp.Fill(0);
                        var ans2 = new string[N, N];
                        for (var i = 0; i < N; ++i)
                        {
                            for (var j = 0; j < N; ++j)
                            {
                                if (fieldItem[i, j] == WALL || fieldItem[i, j] == ENEMY) continue;
                                var nh = dp[i, j];
                                var ope = "S";
                                if (i > 0 && nh.Chmax(dp[i - 1, j])) ope = "M D";
                                if (i < N - 1 && nh.Chmax(dp[i + 1, j])) ope = "M U";
                                if (j > 0 && nh.Chmax(dp[i, j - 1])) ope = "M R";
                                if (j < N - 1 && nh.Chmax(dp[i, j + 1])) ope = "M L";
                                --nh;
                                foreach (var item in effect[i, j])
                                {
                                    if (turn % yxd[item].d == 0)
                                    {
                                        nh -= D;
                                    }
                                }
                                ndp[i, j] = nh;
                                ans2[i, j] = ope;
                            }
                        }
                        var t = dp; dp = ndp; ndp = t;
                        if (maxH.Chmax(dp[goalY, goalX])) finishTurn = turn;
                        ansField.Add(ans2);
                    }

                    var firstAnswer = new List<string>();
                    {
                        var nowx = goalX;
                        var nowy = goalY;
                        for (var turn = finishTurn - ans.turn - 1; turn >= 0; --turn)
                        {
                            var ope = ansField[turn][nowy, nowx];
                            firstAnswer.Add(ope);
                            if (ope == "M D") --nowy;
                            else if (ope == "M U") ++nowy;
                            else if (ope == "M R") --nowx;
                            else if (ope == "M L") ++nowx;
                        }
                        firstAnswer.Reverse();
                    }

                    foreach (var item in ans.Item3)
                    {
                        Console.WriteLine(item);
                    }
                    foreach (var item in firstAnswer)
                    {
                        Console.WriteLine(item);
                    }
                }
            }
            else
            {
                var dp = new int[N, N];
                var ndp = new int[N, N];
                var ansField = new List<string[,]>();
                dp[startY, startX] = H;
                var finishTurn = 0;
                var maxH = 0;
                for (var turn = 1; turn < 300; ++turn)
                {
                    ndp.Fill(0);
                    var ans = new string[N, N];
                    for (var i = 0; i < N; ++i)
                    {
                        for (var j = 0; j < N; ++j)
                        {
                            if (fieldItem[i, j] == WALL || fieldItem[i, j] == ENEMY) continue;
                            var nh = dp[i, j];
                            var ope = "S";
                            if (i > 0 && nh.Chmax(dp[i - 1, j])) ope = "M D";
                            if (i < N - 1 && nh.Chmax(dp[i + 1, j])) ope = "M U";
                            if (j > 0 && nh.Chmax(dp[i, j - 1])) ope = "M R";
                            if (j < N - 1 && nh.Chmax(dp[i, j + 1])) ope = "M L";
                            --nh;
                            foreach (var item in effect[i, j])
                            {
                                if (turn % yxd[item].d == 0)
                                {
                                    nh -= D;
                                }
                            }
                            ndp[i, j] = nh;
                            ans[i, j] = ope;
                        }
                    }
                    var t = dp; dp = ndp; ndp = t;
                    if (maxH.Chmax(dp[keyY, keyX])) finishTurn = turn;
                    ansField.Add(ans);
                }
                var firstAnswer = new List<string>();
                {
                    var nowx = keyX;
                    var nowy = keyY;
                    for (var turn = finishTurn - 1; turn >= 0; --turn)
                    {
                        var ope = ansField[turn][nowy, nowx];
                        firstAnswer.Add(ope);
                        if (ope == "M D") --nowy;
                        else if (ope == "M U") ++nowy;
                        else if (ope == "M R") --nowx;
                        else if (ope == "M L") ++nowx;
                        if (fieldItem[nowy, nowx] <= JEWEL) fieldItem[nowy, nowx] = 0;
                    }
                    firstAnswer.Reverse();
                }
                var pool1 = new List<LIB_Bitset>(10000);
                var pool2 = new List<LIB_Bitset>(10000);

                Func<int, int, double, bool, (int lastHealth, int count, string[])> calc = (startTurn, startHelath, gain, dry) =>
                {
                    var dp = new (double health, int count, LIB_Bitset got)[N, N];
                    var ndp = new (double health, int count, LIB_Bitset got)[N, N];
                    var ansField = new List<string[,]>();
                    dp[keyY, keyX] = (startHelath, 0, new LIB_Bitset(jewelNum + 1));
                    var maxH = 0.0;
                    var maxCount = 0;
                    var finishTurn = 0;
                    for (var turn = startTurn; turn < 1000; ++turn)
                    {
                        ndp.Fill((0, 0, null));
                        string[,] ans = null;
                        if (!dry) ans = new string[N, N];
                        for (var i = 0; i < N; ++i)
                        {
                            for (var j = 0; j < N; ++j)
                            {
                                if (fieldItem[i, j] == WALL || fieldItem[i, j] == ENEMY) continue;
                                var nh = dp[i, j].health;
                                var ope = "S";
                                var count = dp[i, j].count;
                                var got = dp[i, j].got;
                                if (!(got is null)) pool2.Add(got);
                                if (i > 0 && nh.Chmax(dp[i - 1, j].health))
                                {
                                    ope = "M D";
                                    count = dp[i - 1, j].count;
                                    got = dp[i - 1, j].got;
                                }
                                if (i < N - 1 && nh.Chmax(dp[i + 1, j].health))
                                {
                                    ope = "M U";
                                    count = dp[i + 1, j].count;
                                    got = dp[i + 1, j].got;
                                }
                                if (j > 0 && nh.Chmax(dp[i, j - 1].health))
                                {
                                    ope = "M R";
                                    count = dp[i, j - 1].count;
                                    got = dp[i, j - 1].got;
                                }
                                if (j < N - 1 && nh.Chmax(dp[i, j + 1].health))
                                {
                                    ope = "M L";
                                    count = dp[i, j + 1].count;
                                    got = dp[i, j + 1].got;
                                }
                                nh -= 1;
                                foreach (var item in effect[i, j])
                                {
                                    if (turn % yxd[item].d == 0) nh -= D;
                                }
                                if (nh <= 0) continue;
                                if (pool1.Count > 0)
                                {
                                    var nextgot = pool1[pool1.Count - 1];
                                    pool1.RemoveAt(pool1.Count - 1);
                                    for (var k = 0; k < nextgot.ary.Length; ++k)
                                    {
                                        nextgot.ary[k] = got.ary[k];
                                    }
                                    got = nextgot;
                                }
                                else
                                {
                                    got = got | 0;
                                }
                                if (0 < fieldItem[i, j] && fieldItem[i, j] <= JEWEL && !got[fieldItem[i, j]])
                                {
                                    nh += gain;
                                    ++count;
                                    got[fieldItem[i, j]] = true;
                                }
                                ndp[i, j] = (nh, count, got);
                                if (!dry) ans[i, j] = ope;
                            }
                        }
                        var t = dp; dp = ndp; ndp = t;
                        var t3 = pool1; pool1 = pool2; pool2 = t3;

                        if (maxH.Chmax(dp[goalY, goalX].health))
                        {
                            maxCount = dp[goalY, goalX].count;
                            finishTurn = ansField.Count;
                        }
                        ansField.Add(ans);
                    }
                    var firstAnswer = new List<string>();
                    if (!dry)
                    {
                        var nowx = goalX;
                        var nowy = goalY;
                        for (var turn = finishTurn; turn >= 0; --turn)
                        {
                            var ope = ansField[turn][nowy, nowx];
                            firstAnswer.Add(ope);
                            if (ope == "M D") --nowy;
                            else if (ope == "M U") ++nowy;
                            else if (ope == "M R") --nowx;
                            else if (ope == "M L") ++nowx;
                        }
                        firstAnswer.Reverse();
                    }
                    return ((int)maxH, maxCount, firstAnswer.ToArray());
                };
                {
                    var minokgain = double.MaxValue;
                    var gain = 0.0;
                    while (true)
                    {
                        var turnTime = DateTime.Now;
                        var mid = (xorshift % 2000) / 1000.0 + 9.0;
                        var res = calc(finishTurn + 1, maxH, mid, true);
                        var hp = res.lastHealth - res.count * mid;
                        if (hp >= 0)
                        {
                            if (minokgain.Chmin(hp)) gain = mid;
                        }
                        var elapsed = (DateTime.Now - turnTime).TotalMilliseconds;
                        if ((DateTime.Now - startTime).TotalMilliseconds + elapsed * 2 > 2900) break;
                    }

                    var ans = calc(finishTurn + 1, maxH, gain, false);
                    foreach (var item in firstAnswer)
                    {
                        Console.WriteLine(item);
                    }
                    foreach (var item in ans.Item3)
                    {
                        Console.WriteLine(item);
                    }
                }
            }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static string[] args;
        static public void Main(string[] args_t) { args = args_t; if (args_t.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
        static bool Chmax<T>(this ref T lhs, T rhs) where T : struct, IComparable<T> { if (lhs.CompareTo(rhs) < 0) { lhs = rhs; return true; } return false; }
        static bool Chmin<T>(this ref T lhs, T rhs) where T : struct, IComparable<T> { if (lhs.CompareTo(rhs) > 0) { lhs = rhs; return true; } return false; }
        static void Fill<T>(this T[] array, T value) => array.AsSpan().Fill(value);
        static void Fill<T>(this T[,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0], array.Length).Fill(value);
        static void Fill<T>(this T[,,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0, 0], array.Length).Fill(value);
        static void Fill<T>(this T[,,,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0, 0, 0], array.Length).Fill(value);
    }
}
namespace Library {
    class LIB_BitUtil
    {
        public static readonly ulong[] BitMask;
        static LIB_BitUtil()
        {
            BitMask = new ulong[64];
            BitMask[0] = 1;
            for (var i = 1; i < 64; i++) BitMask[i] = BitMask[i - 1] << 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> ScanOne(long value)
        {
            for (; value > 0; value &= value - 1) yield return value & -value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public int MSB(long value)
        {
            return 64 - (int)System.Runtime.Intrinsics.X86.Lzcnt.X64.LeadingZeroCount((ulong)value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public int LSB(long value)
        {
            return 64 - (int)System.Runtime.Intrinsics.X86.Lzcnt.X64.LeadingZeroCount((ulong)(value & -value));
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long PopCount(ulong value)
        {
            return (long)System.Runtime.Intrinsics.X86.Popcnt.X64.PopCount(value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long PopCount(long value) => PopCount((ulong)value);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(ulong value, int idx) => (value & BitMask[idx]) != 0;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(long value, int idx) => IsSet((ulong)value, idx);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public T[] BitZentansaku<T>(long bitlen, T e, Func<int, T, T> f)
        {
            var retlen = 1 << (int)bitlen;
            var ret = new T[retlen];
            var idx = 0;
            ret[0] = e;
            for (var i = 1; i < retlen; ++i)
            {
                if (i == (1 << (idx + 1))) ++idx;
                ret[i] = f(idx, ret[i & (~(1 << idx))]);
            }
            return ret;
        }
    }
    class LIB_Bitset : IEquatable<LIB_Bitset>
    {
        long n;
        public ulong[] ary;
        static readonly ulong[] ceil;
        static LIB_Bitset()
        {
            ceil = new ulong[64];
            ceil[0] = 0xffffffffffffffff;
            ceil[1] = 1;
            for (var i = 2; i < 64; i++)
            {
                ceil[i] = (ceil[i - 1] << 1) | 1;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Bitset(long size)
        {
            if (size <= 0) throw new Exception();
            n = size;
            ary = new ulong[((n - 1) >> 6) + 1];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Bitset(LIB_Bitset src)
        {
            n = src.n;
            ary = src.ary.ToArray();
        }
        public long Count => n;
        public long PopCount => ary.Sum(e => LIB_BitUtil.PopCount(e));
        public bool this[int idx]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return LIB_BitUtil.IsSet(ary[idx >> 6], idx & 63); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            set
            {
                if (value) ary[idx >> 6] |= LIB_BitUtil.BitMask[idx & 63];
                else ary[idx >> 6] &= ~LIB_BitUtil.BitMask[idx & 63];
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator &(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] &= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator |(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] |= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void OrEqual(LIB_Bitset y)
        {
            if (n < y.n)
            {
                n = y.n;
                var tmp = new ulong[y.n];
                for (var i = 0; i < ary.Length; i++) tmp[i] = ary[i];
                ary = tmp;
            }
            for (var i = 0; i < y.ary.Length; i++) ary[i] |= y.ary[i];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ^(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] ^= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator &(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] &= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator |(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] |= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ^(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] ^= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Flip()
        {
            for (var i = 0; i < ary.Length; i++) ary[i] = ~ary[i];
            ary[ary.Length - 1] &= ceil[n & 63];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ~(LIB_Bitset x)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = ~x.ary[i];
            ret.ary[ret.ary.Length - 1] &= ceil[ret.n & 63];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void ShiftLeft(int num)
        {
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var i = ary.Length - 1;
            for (; i >= moveCnt; i--)
            {
                ary[i] = ary[i - moveCnt] << moveBit;
                if (moveBit > 0 && i > moveCnt) ary[i] |= ary[i - moveCnt - 1] >> (64 - moveBit);
            }
            for (; i >= 0; --i) ary[i] = 0;
            ary[ary.Length - 1] &= ceil[n & 63];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator <<(LIB_Bitset x, int num)
        {
            var ret = new LIB_Bitset(x.n);
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var i = ret.ary.Length - 1;
            for (; i >= moveCnt; i--)
            {
                ret.ary[i] = x.ary[i - moveCnt] << moveBit;
                if (moveBit > 0 && i > moveCnt) ret.ary[i] |= x.ary[i - moveCnt - 1] >> (64 - moveBit);
            }
            for (; i >= 0; --i) ret.ary[i] = 0;
            ret.ary[ret.ary.Length - 1] &= ceil[ret.n & 63];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void ShiftRight(int num)
        {
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var aryMax = ary.Length - moveCnt;
            var i = 0;
            for (; i < aryMax; i++)
            {
                ary[i] = ary[i + moveCnt] >> moveBit;
                if (moveBit > 0 && i < aryMax - 1) ary[i] |= ary[i + moveCnt + 1] << (64 - moveBit);
            }
            for (; i < ary.Length; ++i) ary[i] = 0;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator >>(LIB_Bitset x, int num)
        {
            var ret = new LIB_Bitset(x.n);
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var aryMax = ret.ary.Length - moveCnt;
            var i = 0;
            for (; i < aryMax; i++)
            {
                ret.ary[i] = x.ary[i + moveCnt] >> moveBit;
                if (moveBit > 0 && i < aryMax - 1) ret.ary[i] |= x.ary[i + moveCnt + 1] << (64 - moveBit);
            }
            for (; i < ret.ary.Length; ++i) ret.ary[i] = 0;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator ==(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var i = 0;
            for (; i < y.ary.Length; i++) if (x.ary[i] != y.ary[i]) return false;
            for (; i < x.ary.Length; i++) if (x.ary[i] != 0) return false;
            return true;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator !=(LIB_Bitset x, LIB_Bitset y) => !(x == y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(LIB_Bitset x) => x == this;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object x) => x == null ? false : Equals((LIB_Bitset)x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode()
        {
            var t = ary.Aggregate((a, x) => a ^ x);
            return (int)(((t >> 32) ^ t) & 0x000000007fffffff);
        }
    }
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[2048];
        int len, ptr;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 2048)) <= 0)
                {
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = (ret << 3) + (ret << 1) + b - '0';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
}