using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Text;
using System.Threading.Tasks;

namespace YukiCoder
{
	class Program
	{
		[MethodImpl(MethodImplOptions.AggressiveOptimization)]
		static void Main()
		{
#if DEBUG
			object locker = new object();
			long sum = 0;
			int size = 100;

			bool parallel = true;
			if (parallel) {
				Parallel.For(0, size, i => {
					var score = Run(i);
					lock (locker) {
						Console.WriteLine($"{i:d4}: {score:d6}");
						Console.Out.Flush();
						sum += score;
					}
				});
			} else {
				for (int i = 0; i < size; i++) {
					var score = Run(i);
					lock (locker) {
						Console.WriteLine($"{i:d4}: {score:d6}");
						Console.Out.Flush();
						sum += score;
					}
				}
			}

			sum /= size / 100;
			Console.WriteLine("");
			Console.WriteLine($"sum: {sum}");
			Console.Out.Flush();
#else
			Run(-1);
#endif
		}

		private static long Run(int number)
		{
			var sw = new Stopwatch();
			sw.Start();

			bool isDebug = false;
#if DEBUG
			using var cin = number >= 0
				? new Scanner(@$"D:\yukico\score002\input\{number:d4}.txt")
				: new Scanner();
			isDebug = true;
#else
			using var cin = new Scanner();
#endif

			var (n, m) = cin.Int2();
			var planets = new (int x, int y)[n];
			for (int i = 0; i < n; i++) {
				planets[i] = cin.Int2();
			}

			var stations = new (int x, int y)[m];
		
			var root = new List<(int t, int i)>();
			for (int i = 0; i < n; i++) {
				root.Add((1, i));
			}

			//root.Add((1, 1));

			var opt = new TwoOpt<(int t, int i)>(
				root.AsSpan(),
				true,
				(x, y) => CalculateOne(x, y, planets, stations));
			opt.InitializeByNearestNeighbour();
			while (sw.ElapsedMilliseconds < 400) {
				opt.Optimize();
			}

			root = opt.Array.ToList();

			for (int i = 0; i < m; i++) {
				int mi = 0;
				long max = 0;
				int xxx = 0;
				int yyy = 0;
				var addList = new HashSet<int>();
				for (int l = 1; l < 20; l++) {
					for (int j = 0; j < root.Count - 1; j++) {
						if (root[j].t == 2 || root[j + 1].t == 2) {
							continue;
						}

						int ll = Math.Min(l, root.Count - 1 - j + 1);
						int xx = 0;
						int yy = 0;
						for (int tt = 0; tt < ll; tt++) {
							xx += planets[root[j + tt].i].x;
							yy += planets[root[j + tt].i].y;
						}

						xx /= ll;
						yy /= ll;
						stations[i] = (xx, yy);

						long del = 0;
						var tempAddList = new HashSet<int>();
						for (int k = 0; k < root.Count - 1; k++) {
							var before = CalculateOne(root[k], root[k + 1], planets, stations);
							var after = CalculateOne(root[k], (2, i), planets, stations)
								+ CalculateOne((2, i), root[k + 1], planets, stations);
							if (after < before) {
								tempAddList.Add(k);
								del += before - after;
							}
						}

						if (del > max) {
							mi = j;
							max = del;
							addList = tempAddList;
							xxx = xx;
							yyy = yy;
						}
					}
				}
				
				stations[i] = (xxx, yyy);

				var nextRoot = new List<(int t, int i)>();
				for (int j = 0; j < root.Count; j++) {
					nextRoot.Add(root[j]);
					if (addList.Contains(j)) {
						nextRoot.Add((2, i));
					}
				}

				root = nextRoot;
			}

			for (int i = 0; i < n; i++) {
				var addList = new HashSet<int>();
				for (int j = 0; j < root.Count - 1; j++) {
					var before = CalculateOne(root[j], root[j + 1], planets, stations);
					var after = CalculateOne(root[j], (1, i), planets, stations)
						+ CalculateOne((1, i), root[j + 1], planets, stations);
					if (after < before) {
						addList.Add(j);
					}
				}

				if (addList.Count > 0) {
					var nextRoot = new List<(int t, int i)>();
					for (int j = 0; j < root.Count; j++) {
						nextRoot.Add(root[j]);
						if (addList.Contains(j)) {
							nextRoot.Add((1, i));
						}
					}

					root = nextRoot;
				}
			}

			long score = Calculate(root, planets, stations);
			var rnd = new Random();
			while (sw.ElapsedMilliseconds < 920) {
				int tm = rnd.Next(0, m);
				int dd = rnd.Next(0, 4);

				var (ox, oy) = stations[tm];
				if ((dd & 1) == 0) {
					if ((dd & 2) == 0) {
						stations[tm] = (ox - 1, oy);
					} else {
						stations[tm] = (ox + 1, oy);
					}
				} else {
					if ((dd & 2) == 0) {
						stations[tm] = (ox, oy - 1);
					} else {
						stations[tm] = (ox, oy + 1);
					}
				}

				var tempScore = Calculate(root, planets, stations);
				if (tempScore >= score) {
					score = tempScore;
				} else {
					stations[tm] = (ox, oy);
				}
			}

			if (isDebug == false) {
				for (int i = 0; i < m; i++) {
					Console.WriteLine($"{stations[i].x} {stations[i].y}");
				}

				Console.WriteLine(root.Count);
				foreach (var (t, i) in root) {
					Console.WriteLine($"{t} {i + 1}");
				}
			} else {
				var sb = new StringBuilder();
				for (int i = 0; i < m; i++) {
					sb.AppendLine($"{stations[i].x} {stations[i].y}");
				}

				sb.AppendLine(root.Count.ToString());
				foreach (var (t, i) in root) {
					sb.AppendLine($"{t} {i + 1}");
				}

				cin.Save(sb.ToString());
			}

			return score;
		}

		private static long CalculateOne(
			(int t, int i) r1,
			(int t, int i) r2,
			(int x, int y)[] platents,
			(int x, int y)[] stations)
		{
			var (t1, i1) = r1;
			var (t2, i2) = r2;
			var (x1, y1) = t1 == 1 ? platents[i1] : stations[i1];
			var (x2, y2) = t2 == 1 ? platents[i2] : stations[i2];

			long dx = x1 - x2;
			long dy = y1 - y2;
			long d = dx * dx + dy * dy;
			long alpha = t1 != t2
				? 5
				: t1 == 1 ? 25 : 1;
			return d * alpha;
		}

		private static long Calculate(
			List<(int t, int i)> root,
			(int x, int y)[] planets,
			(int x, int y)[] stations)
		{
			long s = 0;
			int k = root.Count;
			for (int i = 0; i < k; i++) {
				long temp = CalculateOne(root[i], root[(i + 1) % k], planets, stations);
				s += temp;
			}

			long score = (long)Math.Round(1000000000L / (1000 + Math.Sqrt(s)));
			return score;
		}
	}

	public class TwoOpt<T>
	{
		private readonly int _n;
		private readonly int _m;
		private readonly T[] _array;
		private readonly Func<T, T, long> _getCost;
		private readonly Random _rnd = new Random();

		public T this[int i] => _array[i];
		public T[] Array => _array;

		public TwoOpt(
			ReadOnlySpan<T> src,
			bool isRing,
			Func<T, T, long> getCost)
		{
			_n = src.Length;
			_getCost = getCost;

			_m = isRing ? _n + 1 : _n;
			_array = new T[_m];
			for (int i = 0; i < _n; i++) {
				_array[i] = src[i];
			}

			if (isRing) {
				_array[_n] = _array[0];
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void InitializeByNearestNeighbour() => InitializeByNearestNeighbour(0, _n);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void InitializeByNearestNeighbour(int left, int right)
		{
			for (int i = left + 1; i < right; i++) {
				long min = long.MaxValue;
				int minIndex = 0;
				for (int j = i; j < right; j++) {
					long cost = _getCost(_array[i - 1], _array[j]);
					if (cost < min) {
						min = cost;
						minIndex = j;
					}
				}

				(_array[minIndex], _array[i]) = (_array[i], _array[minIndex]);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Optimize() => Optimize(1, 0, _m);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Optimize(int count) => Optimize(count, 0, _m);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Optimize(int count, int left, int right)
		{
			for (int k = 0; k < count; k++) {
				int i = _rnd.Next(left, right - 1);
				int j = _rnd.Next(left + 1, right);
				while (i + 1 == j) {
					j = _rnd.Next(left + 1, right);
				}

				if (j < i) {
					(i, j) = (j, i);
				}

				long before = _getCost(_array[i], _array[i + 1])
					+ _getCost(_array[j - 1], _array[j]);
				long after = _getCost(_array[i], _array[j - 1])
					+ _getCost(_array[i + 1], _array[j]);
				if (after < before) {
					++i;
					--j;
					while (i < j) {
						(_array[i], _array[j]) = (_array[j], _array[i]);
						++i;
						--j;
					}
				}
			}
		}

		public (long delta, int targetIndex) EstimateExchange(int removeIndex, T newValue)
			=> EstimateExchange(removeIndex, newValue, 1, _m - 1);
		public (long delta, int targetIndex) EstimateExchange(
			int removeIndex, T newValue, int left, int right)
		{
			long after = long.MaxValue;
			int target = 1;
			for (int i = left; i <= right; i++) {
				int l = i - 1;
				if (l == removeIndex) {
					l--;
				}

				int r = i;
				if (r == removeIndex) {
					r++;
				}

				long tempAfter = _getCost(_array[l], newValue) + _getCost(newValue, _array[r]);
				if (tempAfter < after) {
					after = tempAfter;
					target = i;
				}
			}

			if (removeIndex == target || removeIndex == target - 1) {
				long before = _getCost(_array[removeIndex - 1], _array[removeIndex])
					+ _getCost(_array[removeIndex], _array[removeIndex + 1]);
				return (after - before, target);
			} else {
				long delta = 0;
				delta -= _getCost(_array[removeIndex - 1], _array[removeIndex]);
				delta -= _getCost(_array[removeIndex], _array[removeIndex + 1]);
				delta += _getCost(_array[removeIndex - 1], _array[removeIndex + 1]);
				delta -= _getCost(_array[target - 1], _array[target]);
				delta += after;
				return (delta, target);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Exchange(int removeIndex, T newValue, bool forced)
			=> Exchange(removeIndex, newValue, 1, _m - 1, forced);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Exchange(int removeIndex, T newValue, int left, int right, bool forced)
		{
			var (delta, target) = EstimateExchange(removeIndex, newValue, left, right);
			if (forced || delta < 0) {
				Exchange(removeIndex, target, newValue);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Exchange(int removeIndex, int insertIndex, T newValue)
		{
			if (removeIndex < insertIndex) {
				for (int i = removeIndex; i < insertIndex - 1; i++) {
					_array[i] = _array[i + 1];
				}

				_array[insertIndex - 1] = newValue;
			} else if (removeIndex > insertIndex) {
				for (int i = removeIndex - 1; i >= insertIndex; i--) {
					_array[i + 1] = _array[i];
				}

				_array[insertIndex] = newValue;
			} else {
				_array[removeIndex] = newValue;
			}
		}
	}

	public struct BitFlag
	{
		public static BitFlag Begin() => 0;
		public static BitFlag End(int bitCount) => 1 << bitCount;
		public static BitFlag FromBit(int bitNumber) => 1 << bitNumber;
		public static BitFlag Fill(int count) => (1 << count) - 1;

		public static IEnumerable<BitFlag> All(int n)
		{
			for (var f = Begin(); f < End(n); ++f) {
				yield return f;
			}
		}

		private readonly int flags_;
		public int Flag => flags_;
		public bool this[int bitNumber] => (flags_ & (1 << bitNumber)) != 0;
		public BitFlag(int flags) { flags_ = flags; }

		public bool Has(BitFlag target) => (flags_ & target.flags_) == target.flags_;
		public bool Has(int target) => (flags_ & target) == target;
		public bool HasBit(int bitNumber) => (flags_ & (1 << bitNumber)) != 0;
		public BitFlag OrBit(int bitNumber) => flags_ | (1 << bitNumber);
		public BitFlag AndBit(int bitNumber) => flags_ & (1 << bitNumber);
		public BitFlag XorBit(int bitNumber) => flags_ ^ (1 << bitNumber);
		public BitFlag ComplementOf(BitFlag sub) => flags_ ^ sub.flags_;
		public int PopCount() => BitOperations.PopCount((uint)flags_);

		public static BitFlag operator ++(BitFlag src) => new BitFlag(src.flags_ + 1);
		public static BitFlag operator --(BitFlag src) => new BitFlag(src.flags_ - 1);
		public static BitFlag operator |(BitFlag lhs, BitFlag rhs)
			=> new BitFlag(lhs.flags_ | rhs.flags_);
		public static BitFlag operator |(BitFlag lhs, int rhs)
			=> new BitFlag(lhs.flags_ | rhs);
		public static BitFlag operator |(int lhs, BitFlag rhs)
			=> new BitFlag(lhs | rhs.flags_);
		public static BitFlag operator &(BitFlag lhs, BitFlag rhs)
			=> new BitFlag(lhs.flags_ & rhs.flags_);
		public static BitFlag operator &(BitFlag lhs, int rhs)
			=> new BitFlag(lhs.flags_ & rhs);
		public static BitFlag operator &(int lhs, BitFlag rhs)
			=> new BitFlag(lhs & rhs.flags_);
		public static BitFlag operator ^(BitFlag lhs, BitFlag rhs)
			=> new BitFlag(lhs.flags_ ^ rhs.flags_);
		public static BitFlag operator ^(BitFlag lhs, int rhs)
			=> new BitFlag(lhs.flags_ ^ rhs);
		public static BitFlag operator ^(int lhs, BitFlag rhs)
			=> new BitFlag(lhs ^ rhs.flags_);
		public static BitFlag operator <<(BitFlag bit, int shift) => bit.flags_ << shift;
		public static BitFlag operator >>(BitFlag bit, int shift) => bit.flags_ >> shift;

		public static bool operator <(BitFlag lhs, BitFlag rhs) => lhs.flags_ < rhs.flags_;
		public static bool operator <(BitFlag lhs, int rhs) => lhs.flags_ < rhs;
		public static bool operator <(int lhs, BitFlag rhs) => lhs < rhs.flags_;
		public static bool operator >(BitFlag lhs, BitFlag rhs) => lhs.flags_ > rhs.flags_;
		public static bool operator >(BitFlag lhs, int rhs) => lhs.flags_ > rhs;
		public static bool operator >(int lhs, BitFlag rhs) => lhs > rhs.flags_;
		public static bool operator <=(BitFlag lhs, BitFlag rhs) => lhs.flags_ <= rhs.flags_;
		public static bool operator <=(BitFlag lhs, int rhs) => lhs.flags_ <= rhs;
		public static bool operator <=(int lhs, BitFlag rhs) => lhs <= rhs.flags_;
		public static bool operator >=(BitFlag lhs, BitFlag rhs) => lhs.flags_ >= rhs.flags_;
		public static bool operator >=(BitFlag lhs, int rhs) => lhs.flags_ >= rhs;
		public static bool operator >=(int lhs, BitFlag rhs) => lhs >= rhs.flags_;

		public static implicit operator BitFlag(int t) => new BitFlag(t);
		public static implicit operator int(BitFlag t) => t.flags_;

		public override string ToString() => $"{Convert.ToString(flags_, 2).PadLeft(32, '0')} ({flags_})";

		public SubBitsEnumerator SubBits => new SubBitsEnumerator(flags_);
		public struct SubBitsEnumerator : IEnumerable<BitFlag>
		{
			private readonly int flags_;
			public SubBitsEnumerator(int flags)
			{
				flags_ = flags;
			}

			IEnumerator<BitFlag> IEnumerable<BitFlag>.GetEnumerator() => new Enumerator(flags_);
			IEnumerator IEnumerable.GetEnumerator() => new Enumerator(flags_);
			public Enumerator GetEnumerator() => new Enumerator(flags_);
			public struct Enumerator : IEnumerator<BitFlag>
			{
				private readonly int src_;
				public BitFlag Current { get; private set; }
				object IEnumerator.Current => Current;

				public Enumerator(int flags)
				{
					src_ = flags;
					Current = flags + 1;
				}

				public void Dispose() { }
				[MethodImpl(MethodImplOptions.AggressiveInlining)]
				public bool MoveNext() => (Current = --Current & src_) > 0;
				[MethodImpl(MethodImplOptions.AggressiveInlining)]
				public void Reset() => Current = src_;
			}
		}
	}

	public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
	{
		public static HashMap<TKey, TValue> Merge(
			HashMap<TKey, TValue> src1,
			HashMap<TKey, TValue> src2,
			Func<TValue, TValue, TValue> mergeValues)
		{
			if (src1.Count < src2.Count) {
				(src1, src2) = (src2, src1);
			}

			foreach (var key in src2.Keys) {
				src1[key] = mergeValues(src1[key], src2[key]);
			}

			return src1;
		}

		private readonly Func<TKey, TValue> initialzier_;
		public HashMap(Func<TKey, TValue> initialzier)
			: base()
		{
			initialzier_ = initialzier;
		}

		public HashMap(Func<TKey, TValue> initialzier, int capacity)
			: base(capacity)
		{
			initialzier_ = initialzier;
		}

		new public TValue this[TKey key]
		{
			get
			{
				if (TryGetValue(key, out TValue value)) {
					return value;
				} else {
					var init = initialzier_(key);
					base[key] = init;
					return init;
				}
			}

			set { base[key] = value; }
		}

		public HashMap<TKey, TValue> Merge(
			HashMap<TKey, TValue> src,
			Func<TValue, TValue, TValue> mergeValues)
		{
			foreach (var key in src.Keys) {
				this[key] = mergeValues(this[key], src[key]);
			}

			return this;
		}
	}

	public class JagList2<T> where T : struct
	{
		private readonly int n_;
		private readonly List<T>[] tempValues_;
		private T[][] values_;

		public int Count => n_;
		public List<T>[] Raw => tempValues_;
		public T[][] Values => values_;
		public T[] this[int index] => values_[index];

		public JagList2(int n)
		{
			n_ = n;
			tempValues_ = new List<T>[n];
			for (int i = 0; i < n; ++i) {
				tempValues_[i] = new List<T>();
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Add(int i, T value) => tempValues_[i].Add(value);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Build()
		{
			values_ = new T[n_][];
			for (int i = 0; i < values_.Length; ++i) {
				values_[i] = tempValues_[i].ToArray();
			}
		}
	}

	public class DijkstraQ
	{
		private int count_ = 0;
		private long[] distanceHeap_;
		private int[] vertexHeap_;

		public int Count => count_;
		public DijkstraQ()
		{
			distanceHeap_ = new long[8];
			vertexHeap_ = new int[8];
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Enqueue(long distance, int v)
		{
			if (distanceHeap_.Length == count_) {
				var newDistanceHeap = new long[distanceHeap_.Length << 1];
				var newVertexHeap = new int[vertexHeap_.Length << 1];
				Unsafe.CopyBlock(
					ref Unsafe.As<long, byte>(ref newDistanceHeap[0]),
					ref Unsafe.As<long, byte>(ref distanceHeap_[0]),
					(uint)(8 * count_));
				Unsafe.CopyBlock(
					ref Unsafe.As<int, byte>(ref newVertexHeap[0]),
					ref Unsafe.As<int, byte>(ref vertexHeap_[0]),
					(uint)(4 * count_));
				distanceHeap_ = newDistanceHeap;
				vertexHeap_ = newVertexHeap;
			}

			ref var dRef = ref distanceHeap_[0];
			ref var vRef = ref vertexHeap_[0];
			Unsafe.Add(ref dRef, count_) = distance;
			Unsafe.Add(ref vRef, count_) = v;
			++count_;

			int c = count_ - 1;
			while (c > 0) {
				int p = (c - 1) >> 1;
				var tempD = Unsafe.Add(ref dRef, p);
				if (tempD <= distance) {
					break;
				} else {
					Unsafe.Add(ref dRef, c) = tempD;
					Unsafe.Add(ref vRef, c) = Unsafe.Add(ref vRef, p);
					c = p;
				}
			}

			Unsafe.Add(ref dRef, c) = distance;
			Unsafe.Add(ref vRef, c) = v;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long distance, int v) Dequeue()
		{
			ref var dRef = ref distanceHeap_[0];
			ref var vRef = ref vertexHeap_[0];
			(long distance, int v) ret = (dRef, vRef);
			int n = count_ - 1;

			var distance = Unsafe.Add(ref dRef, n);
			var vertex = Unsafe.Add(ref vRef, n);
			int p = 0;
			int c = (p << 1) + 1;
			while (c < n) {
				if (c != n - 1 && Unsafe.Add(ref dRef, c + 1) < Unsafe.Add(ref dRef, c)) {
					++c;
				}

				var tempD = Unsafe.Add(ref dRef, c);
				if (distance > tempD) {
					Unsafe.Add(ref dRef, p) = tempD;
					Unsafe.Add(ref vRef, p) = Unsafe.Add(ref vRef, c);
					p = c;
					c = (p << 1) + 1;
				} else {
					break;
				}
			}

			Unsafe.Add(ref dRef, p) = distance;
			Unsafe.Add(ref vRef, p) = vertex;
			--count_;

			return ret;
		}
	}

	public struct ModInt
	{
		//public const long P = 1000000007;
		public const long P = 998244353;
		//public const long P = 2;
		public const long ROOT = 3;

		// (924844033, 5)
		// (998244353, 3)
		// (1012924417, 5)
		// (167772161, 3)
		// (469762049, 3)
		// (1224736769, 3)

		private long value_;

		public static ModInt New(long value, bool mods) => new ModInt(value, mods);
		public ModInt(long value) => value_ = value;
		public ModInt(long value, bool mods)
		{
			if (mods) {
				value %= P;
				if (value < 0) {
					value += P;
				}
			}

			value_ = value;
		}

		public static ModInt operator +(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (lhs.value_ + rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator +(long lhs, ModInt rhs)
		{
			rhs.value_ = (lhs + rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator +(ModInt lhs, long rhs)
		{
			lhs.value_ = (lhs.value_ + rhs) % P;
			return lhs;
		}

		public static ModInt operator -(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs.value_) % P;
			return lhs;
		}
		public static ModInt operator -(long lhs, ModInt rhs)
		{
			rhs.value_ = (P + lhs - rhs.value_) % P;
			return rhs;
		}
		public static ModInt operator -(ModInt lhs, long rhs)
		{
			lhs.value_ = (P + lhs.value_ - rhs) % P;
			return lhs;
		}

		public static ModInt operator *(ModInt lhs, ModInt rhs)
		{
			lhs.value_ = lhs.value_ * rhs.value_ % P;
			return lhs;
		}
		public static ModInt operator *(long lhs, ModInt rhs)
		{
			rhs.value_ = lhs * rhs.value_ % P;
			return rhs;
		}
		public static ModInt operator *(ModInt lhs, long rhs)
		{
			lhs.value_ = lhs.value_ * rhs % P;
			return lhs;
		}

		public static ModInt operator /(ModInt lhs, ModInt rhs)
			=> lhs * Inverse(rhs);

		public static implicit operator ModInt(long n) => new ModInt(n, true);

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static ModInt Inverse(ModInt value) => Pow(value, P - 2);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static ModInt Pow(long value, long k)
		{
			long ret = 1;
			while (k > 0) {
				if ((k & 1) != 0) {
					ret = ret * value % P;
				}

				value = value * value % P;
				k >>= 1;
			}

			return new ModInt(ret);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long ToLong() => value_;
		public override string ToString() => value_.ToString();
	}

	public static class Helper
	{
		public static long INF => 1L << 50;

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T Clamp<T>(this T value, T min, T max) where T : struct, IComparable<T>
		{
			if (value.CompareTo(min) <= 0) {
				return min;
			}

			if (value.CompareTo(max) >= 0) {
				return max;
			}

			return value;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(this ref T target, T value) where T : struct, IComparable<T>
			=> target = target.CompareTo(value) > 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMin<T>(this ref T target, T value, Action<T> onUpdated)
			where T : struct, IComparable<T>
		{
			if (target.CompareTo(value) > 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(this ref T target, T value) where T : struct, IComparable<T>
			=> target = target.CompareTo(value) < 0 ? value : target;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void UpdateMax<T>(this ref T target, T value, Action<T> onUpdated)
			where T : struct, IComparable<T>
		{
			if (target.CompareTo(value) < 0) {
				target = value;
				onUpdated(value);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static long BinarySearchOKNG(long ok, long ng, Func<long, bool> satisfies)
		{
			while (ng - ok > 1) {
				long mid = (ok + ng) / 2;
				if (satisfies(mid)) {
					ok = mid;
				} else {
					ng = mid;
				}
			}

			return ok;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static long BinarySearchNGOK(long ng, long ok, Func<long, bool> satisfies)
		{
			while (ok - ng > 1) {
				long mid = (ok + ng) / 2;
				if (satisfies(mid)) {
					ok = mid;
				} else {
					ng = mid;
				}
			}

			return ok;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Array1<T>(int n, T initialValue) where T : struct
			=> new T[n].Fill(initialValue);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Array1<T>(int n, Func<int, T> initializer)
			=> Enumerable.Range(0, n).Select(x => initializer(x)).ToArray();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Fill<T>(this T[] array, T value)
			where T : struct
		{
			array.AsSpan().Fill(value);
			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, T initialValule) where T : struct
			=> new T[n, m].Fill(initialValule);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Array2<T>(int n, int m, Func<int, int, T> initializer)
		{
			var array = new T[n, m];
			for (int i = 0; i < n; ++i) {
				for (int j = 0; j < m; ++j) {
					array[i, j] = initializer(i, j);
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,] Fill<T>(this T[,] array, T initialValue)
			where T : struct
		{
			MemoryMarshal.CreateSpan<T>(ref array[0, 0], array.Length).Fill(initialValue);
			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static Span<T> AsSpan<T>(this T[,] array, int i)
			=> MemoryMarshal.CreateSpan<T>(ref array[i, 0], array.GetLength(1));

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,] Array3<T>(int n1, int n2, int n3, T initialValue)
			where T : struct
			=> new T[n1, n2, n3].Fill(initialValue);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,] Fill<T>(this T[,,] array, T initialValue)
			where T : struct
		{
			MemoryMarshal.CreateSpan<T>(ref array[0, 0, 0], array.Length).Fill(initialValue);
			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static Span<T> AsSpan<T>(this T[,,] array, int i, int j)
			=> MemoryMarshal.CreateSpan<T>(ref array[i, j, 0], array.GetLength(2));

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T initialValue)
			where T : struct
			=> new T[n1, n2, n3, n4].Fill(initialValue);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,,] Fill<T>(this T[,,,] array, T initialValue)
			where T : struct
		{
			MemoryMarshal.CreateSpan<T>(ref array[0, 0, 0, 0], array.Length).Fill(initialValue);
			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static Span<T> AsSpan<T>(this T[,,,] array, int i, int j, int k)
			=> MemoryMarshal.CreateSpan<T>(ref array[i, j, k, 0], array.GetLength(3));

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Merge<T>(ReadOnlySpan<T> first, ReadOnlySpan<T> second) where T : IComparable<T>
		{
			var ret = new T[first.Length + second.Length];
			int p = 0;
			int q = 0;
			while (p < first.Length || q < second.Length) {
				if (p == first.Length) {
					ret[p + q] = second[q];
					q++;
					continue;
				}

				if (q == second.Length) {
					ret[p + q] = first[p];
					p++;
					continue;
				}

				if (first[p].CompareTo(second[q]) < 0) {
					ret[p + q] = first[p];
					p++;
				} else {
					ret[p + q] = second[q];
					q++;
				}
			}

			return ret;
		}

		private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static IEnumerable<(int i, int j)> Adjacence4(int i, int j, int imax, int jmax)
		{
			for (int dn = 0; dn < 4; ++dn) {
				int d4i = i + delta4_[dn];
				int d4j = j + delta4_[dn + 1];
				if ((uint)d4i < (uint)imax && (uint)d4j < (uint)jmax) {
					yield return (d4i, d4j);
				}
			}
		}

		private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static IEnumerable<(int i, int j)> Adjacence8(int i, int j, int imax, int jmax)
		{
			for (int dn = 0; dn < 8; ++dn) {
				int d8i = i + delta8_[dn];
				int d8j = j + delta8_[dn + 1];
				if ((uint)d8i < (uint)imax && (uint)d8j < (uint)jmax) {
					yield return (d8i, d8j);
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static IEnumerable<int> SubBitsOf(int bit)
		{
			for (int sub = bit; sub > 0; sub = --sub & bit) {
				yield return sub;
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Reverse(string src)
		{
			var chars = src.ToCharArray();
			for (int i = 0, j = chars.Length - 1; i < j; ++i, --j) {
				var tmp = chars[i];
				chars[i] = chars[j];
				chars[j] = tmp;
			}

			return new string(chars);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Exchange(string src, char a, char b)
		{
			var chars = src.ToCharArray();
			for (int i = 0; i < chars.Length; i++) {
				if (chars[i] == a) {
					chars[i] = b;
				} else if (chars[i] == b) {
					chars[i] = a;
				}
			}

			return new string(chars);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void Swap(this string str, int i, int j)
		{
			var span = str.AsWriteableSpan();
			(span[i], span[j]) = (span[j], span[i]);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static char Replace(this string str, int index, char c)
		{
			var span = str.AsWriteableSpan();
			char old = span[index];
			span[index] = c;
			return old;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static Span<char> AsWriteableSpan(this string str)
		{
			var span = str.AsSpan();
			return MemoryMarshal.CreateSpan(ref MemoryMarshal.GetReference(span), span.Length);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string Join<T>(this IEnumerable<T> values, string separator = "")
			=> string.Join(separator, values);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static string JoinNL<T>(this IEnumerable<T> values)
			=> string.Join(Environment.NewLine, values);
	}

	public static class Extensions
	{
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static Span<T> AsSpan<T>(this List<T> list)
		{
			return Unsafe.As<FakeList<T>>(list).Array.AsSpan(0, list.Count);
		}

		private class FakeList<T>
		{
			public T[] Array = null;
		}
	}

	public class Scanner : IDisposable
	{
		private const int BUFFER_SIZE = 1024;
		private const int ASCII_SPACE = 32;
		private const int ASCII_CHAR_BEGIN = 33;
		private const int ASCII_CHAR_END = 126;
		private readonly string filePath_;
		private readonly Stream stream_;
		private readonly byte[] buf_ = new byte[BUFFER_SIZE];
		private int length_ = 0;
		private int index_ = 0;
		private bool isEof_ = false;

		public Scanner(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				stream_ = Console.OpenStandardInput();
			} else {
				filePath_ = file;
				stream_ = new FileStream(file, FileMode.Open);
			}

			Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush = false
			});
		}

		public void Dispose()
		{
			Console.Out.Flush();
			stream_.Dispose();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string NextLine()
		{
			var sb = new StringBuilder();
			for (var b = Char(); b >= ASCII_SPACE && b <= ASCII_CHAR_END; b = (char)Read()) {
				sb.Append(b);
			}

			return sb.ToString();
		}


		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public char Char()
		{
			byte b;
			do {
				b = Read();
			} while (b < ASCII_CHAR_BEGIN || ASCII_CHAR_END < b);

			return (char)b;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string String()
		{
			var sb = new StringBuilder();
			for (var b = Char(); b >= ASCII_CHAR_BEGIN && b <= ASCII_CHAR_END; b = (char)Read()) {
				sb.Append(b);
			}

			return sb.ToString();
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string[] ArrayString(int length)
		{
			var array = new string[length];
			for (int i = 0; i < length; ++i) {
				array[i] = String();
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int() => (int)Long();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int(int offset) => Int() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int) Int2(int offset = 0)
			=> (Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int) Int3(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int, int) Int4(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (int, int, int, int, int) Int5(int offset = 0)
			=> (Int(offset), Int(offset), Int(offset), Int(offset), Int(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int[] ArrayInt(int length, int offset = 0)
		{
			var array = new int[length];
			for (int i = 0; i < length; ++i) {
				array[i] = Int(offset);
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long()
		{
			long ret = 0;
			byte b;
			bool 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 * 10 + b - '0';
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long(long offset) => Long() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long) Long2(long offset = 0)
			=> (Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long) Long3(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long, long) Long4(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (long, long, long, long, long) Long5(long offset = 0)
			=> (Long(offset), Long(offset), Long(offset), Long(offset), Long(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long[] ArrayLong(int length, long offset = 0)
		{
			var array = new long[length];
			for (int i = 0; i < length; ++i) {
				array[i] = Long(offset);
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public BigInteger Big() => new BigInteger(Long());
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public BigInteger Big(long offset) => Big() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (BigInteger, BigInteger) Big2(long offset = 0)
			=> (Big(offset), Big(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (BigInteger, BigInteger, BigInteger) Big3(long offset = 0)
			=> (Big(offset), Big(offset), Big(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (BigInteger, BigInteger, BigInteger, BigInteger) Big4(long offset = 0)
			=> (Big(offset), Big(offset), Big(offset), Big(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (BigInteger, BigInteger, BigInteger, BigInteger, BigInteger) Big5(long offset = 0)
			=> (Big(offset), Big(offset), Big(offset), Big(offset), Big(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public BigInteger[] ArrayBig(int length, long offset = 0)
		{
			var array = new BigInteger[length];
			for (int i = 0; i < length; ++i) {
				array[i] = Big(offset);
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double() => double.Parse(String(), CultureInfo.InvariantCulture);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double(double offset) => Double() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double) Double2(double offset = 0)
			=> (Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double) Double3(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double, double) Double4(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (double, double, double, double, double) Double5(double offset = 0)
			=> (Double(offset), Double(offset), Double(offset), Double(offset), Double(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double[] ArrayDouble(int length, double offset = 0)
		{
			var array = new double[length];
			for (int i = 0; i < length; ++i) {
				array[i] = Double(offset);
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public decimal Decimal() => decimal.Parse(String(), CultureInfo.InvariantCulture);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public decimal Decimal(decimal offset) => Decimal() + offset;
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (decimal, decimal) Decimal2(decimal offset = 0)
			=> (Decimal(offset), Decimal(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (decimal, decimal, decimal) Decimal3(decimal offset = 0)
			=> (Decimal(offset), Decimal(offset), Decimal(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (decimal, decimal, decimal, decimal) Decimal4(decimal offset = 0)
			=> (Decimal(offset), Decimal(offset), Decimal(offset), Decimal(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public (decimal, decimal, decimal, decimal, decimal) Decimal5(decimal offset = 0)
			=> (Decimal(offset), Decimal(offset), Decimal(offset), Decimal(offset), Decimal(offset));
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public decimal[] ArrayDecimal(int length, decimal offset = 0)
		{
			var array = new decimal[length];
			for (int i = 0; i < length; ++i) {
				array[i] = Decimal(offset);
			}

			return array;
		}

		private byte Read()
		{
			if (isEof_) {
				throw new EndOfStreamException();
			}

			if (index_ >= length_) {
				index_ = 0;
				if ((length_ = stream_.Read(buf_, 0, BUFFER_SIZE)) <= 0) {
					isEof_ = true;
					return 0;
				}
			}

			return buf_[index_++];
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}