Problem:
Score : 600 points

Problem Statement

There is a grid of squares with H+1 horizontal rows and W vertical columns.

You will start at one of the squares in the top row and repeat moving one square right or down. However, for each integer i from 1 through H, you cannot move down from the A_i-th, (A_i + 1)-th, \ldots, B_i-th squares from the left in the i-th row from the top.

For each integer k from 1 through H, find the minimum number of moves needed to reach one of the squares in the (k+1)-th row from the top. (The starting square can be chosen individually for each case.) If, starting from any square in the top row, none of the squares in the (k+1)-th row can be reached, print -1 instead.

Constraints

1 \leq H,W \leq 2\times 10^5

1 \leq A_i \leq B_i \leq W

Input

Input is given from Standard Input in the following format:

H W
A_1 B_1
A_2 B_2
:
A_H B_H

Output

Print H lines. The i-th line should contain the answer for the case k=i.

Sample Input 1

4 4
2 4
1 1
2 3
2 4

Sample Output 1

1
3
6
-1

Let (i,j) denote the square at the i-th row from the top and j-th column from the left.

For k=1, we need one move such as (1,1) → (2,1).

For k=2, we need three moves such as (1,1) → (2,1) → (2,2) → (3,2).

For k=3, we need six moves such as (1,1) → (2,1) → (2,2) → (3,2) → (3,3) → (3,4) → (4,4).

For k=4, it is impossible to reach any square in the fifth row from the top.

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Reflection.Metadata;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;

namespace AtCoder
{
	public class Program
	{
		static void Main()
		{
			var cin = new Scanner();
			var (h, w) = cin.Int2();

			int inf = 100000000;
			var src = Helper.Array(w, x => (x, x));
			var seg = new LazySegmentTree<(int start, int end), int?>(
				src,
				(-inf, inf),
				null,
				(x, y) => (x.end - x.start) < (y.end - y.start) ? x : y,
				(x, y, n) => y is null ? x : (y.Value, x.end),
				(x, y) => y is null ? x : y);

			var sb = new StringBuilder();
			for (int i = 0; i < h; i++) {
				var (a, b) = cin.Int2(-1);

				var index = seg.FindRightest(a, b + 1, x => x.start >= 0);
				if (index >= a) {
					var cur = seg[index];
					if (b == w - 1) {
						seg.Update(a, b + 1, -inf);
					} else {
						seg.Update(a, b + 1, -inf);
						var next = seg.Query(b + 1);
						if (next.start < 0) {
							seg.Update(b + 1, cur.start);
						}
					}
				}

				var ans = seg.Query(0, w);
				if (ans.start < 0) {
					sb.AppendLine("-1");
				} else {
					sb.AppendLine((ans.end - ans.start + i + 1).ToString());
				}
			}

			Console.Write(sb.ToString());
		}
	}

	public class LazySegmentTree<TData, TUpdate> : IEnumerable<TData>
	{
		private readonly int n_;
		private readonly TData[] data_;
		private readonly TUpdate[] lazy_;
		private readonly Func<TData, TData, TData> operate_;
		private readonly Func<TData, TUpdate, int, TData> update_;
		private readonly Func<TUpdate, TUpdate, TUpdate> compose_;
		private readonly TData unitData_;
		private readonly TUpdate unitUpdate_;

		public int Count { get; }

		public TData this[int index] => Query(index);

		public LazySegmentTree(
			int count,
			TData unitData,
			TUpdate unitUpdate,
			Func<TData, TData, TData> operate,
			Func<TData, TUpdate, int, TData> update,
			Func<TUpdate, TUpdate, TUpdate> compose)
		{
			Count = count;
			n_ = 1;
			while (n_ < count) {
				n_ <<= 1;
			}

			data_ = new TData[n_ << 1];
			for (int i = 0; i < data_.Length; i++) {
				data_[i] = unitData;
			}

			lazy_ = new TUpdate[n_ << 1];
			for (int i = 0; i < lazy_.Length; i++) {
				lazy_[i] = unitUpdate;
			}

			unitData_ = unitData;
			unitUpdate_ = unitUpdate;
			operate_ = operate;
			update_ = update;
			compose_ = compose;
		}

		public LazySegmentTree(
			IReadOnlyList<TData> src,
			TData unitData,
			TUpdate unitUpdate,
			Func<TData, TData, TData> operate,
			Func<TData, TUpdate, int, TData> update,
			Func<TUpdate, TUpdate, TUpdate> compose)
			: this(src.Count, unitData, unitUpdate, operate, update, compose)
		{
			for (int i = 0; i < src.Count; i++) {
				data_[i + n_] = src[i];
			}

			for (int i = n_ - 1; i > 0; --i) {
				data_[i] = operate_(data_[i << 1], data_[(i << 1) + 1]);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Update(int v, TUpdate value)
			=> Update(v, v + 1, value);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Update(Range range, TUpdate value)
			=> Update(range.Start.Value, range.End.Value, value);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public void Update(int left, int right, TUpdate value)
		{
			UpdateCore(left, right, 1, 0, n_, value);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		private void UpdateCore(int left, int right, int v, int l, int r, TUpdate value)
		{
			if (left <= l && r <= right) {
				data_[v] = update_(data_[v], value, r - l);
				lazy_[v] = compose_(lazy_[v], value);
			} else if (left < r && l < right) {
				int lc = v << 1;
				int rc = (v << 1) + 1;
				int mid = (l + r) / 2;
				UpdateCore(0, n_, lc, l, mid, lazy_[v]);
				UpdateCore(left, right, lc, l, mid, value);
				UpdateCore(0, n_, rc, mid, r, lazy_[v]);
				UpdateCore(left, right, rc, mid, r, value);
				data_[v] = operate_(data_[lc], data_[rc]);
				lazy_[v] = unitUpdate_;
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public TData Query(int v) => Query(v, v + 1);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public TData Query(Range range) => Query(range.Start.Value, range.End.Value);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public TData Query(int left, int right)
		{
			return QueryCore(left, right, 1, 0, n_);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		private TData QueryCore(int left, int right, int v, int l, int r)
		{
			if (left <= l && r <= right) {
				return update_(data_[v], lazy_[v], r - l);
			} else if (r <= left || right <= l) {
				return unitData_;
			} else {
				return update_(
					operate_(
						QueryCore(left, right, v << 1, l, (l + r) / 2),
						QueryCore(left, right, (v << 1) + 1, (l + r) / 2, r)
					),
					lazy_[v],
					r - l
				);
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int FindRightest(Range range, Func<TData, bool> check)
			=> FindRightest(range.Start.Value, range.End.Value, check);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int FindRightest(int left, int right, Func<TData, bool> check)
			=> FindRightestCore(left, right, 1, 0, n_, check);
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		private int FindRightestCore(int left, int right, int v, int l, int r, Func<TData, bool> check)
		{
			int lc = v << 1;
			int rc = (v << 1) + 1;
			int mid = (l + r) / 2;
			if (v < n_) {
				UpdateCore(0, n_, lc, l, mid, lazy_[v]);
				UpdateCore(0, n_, rc, mid, r, lazy_[v]);
			}

			if (check(data_[v]) == false || r <= left || right <= l) {
				return left - 1;
			} else if (v >= n_) {
				return v - n_;
			} else {
				int vr = FindRightestCore(left, right, rc, mid, r, check);
				if (vr != left - 1) {
					return vr;
				} else {
					return FindRightestCore(left, right, lc, l, mid, check);
				}
			}
		}

		private void Evaluate(int v, int l, int r)
		{
			if (lazy_[v].Equals(unitUpdate_)) {
				return;
			}

			if (v < n_) {
				int lc = v << 1;
				int rc = (v << 1) + 1;
				lazy_[lc] = compose_(lazy_[lc], lazy_[v]);
				lazy_[rc] = compose_(lazy_[rc], lazy_[v]);
			}
			data_[v] = update_(data_[v], lazy_[v], r - l);
			lazy_[v] = unitUpdate_;
		}

		public IEnumerator<TData> GetEnumerator()
		{
			for (int i = 0; i < Count; i++) {
				yield return this[i];
			}
		}

		IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
	};

	public class RedBlackTree<T> : ICollection<T>, IReadOnlyCollection<T>
	{
		private readonly Comparison<T> comparer_;
		private readonly bool isMulti_;
		private Node root_;

		public RedBlackTree(bool isMulti = false)
			: this(Comparer<T>.Default, isMulti) { }

		public RedBlackTree(IEnumerable<T> collection, bool isMulti = false)
			: this(collection, Comparer<T>.Default, isMulti) { }

		public RedBlackTree(IComparer<T> comparer, bool isMulti = false)
			: this(comparer.Compare, isMulti) { }
		public RedBlackTree(Comparison<T> comparer, bool isMulti = false)
		{
			comparer_ = comparer;
			isMulti_ = isMulti;
		}

		public RedBlackTree(IEnumerable<T> collection, IComparer<T> comparer, bool isMulti = false)
			: this(collection, comparer.Compare, isMulti) { }
		public RedBlackTree(IEnumerable<T> collection, Comparison<T> comparer, bool isMulti = false)
		{
			comparer_ = comparer;
			isMulti_ = isMulti;
			var arr = InitializeArray(collection);
			root_ = ConstructRootFromSortedArray(arr, 0, arr.Length - 1, null);
		}

		public int Count => CountOf(root_);
		public T Min
		{
			get
			{
				if (root_ is null) {
					return default;
				}

				var cur = root_;
				while (cur.Left is null == false) {
					cur = cur.Left;
				}

				return cur.Item;
			}
		}

		public T Max
		{
			get
			{
				if (root_ is null) {
					return default;
				}

				var cur = root_;
				while (cur.Right is null == false) {
					cur = cur.Right;
				}

				return cur.Item;
			}
		}

		public bool Contains(T item) => FindNode(item) is null == false;
		public Node FindNode(T item)
		{
			Node current = root_;
			while (current is null == false) {
				int cmp = comparer_(item, current.Item);
				if (cmp == 0) {
					return current;
				}

				current = cmp < 0 ? current.Left : current.Right;
			}

			return null;
		}

		public int Index(Node node)
		{
			var ret = CountOf(node.Left);
			Node prev = node;
			node = node.Parent;
			while (prev != root_) {
				if (node.Left != prev) {
					ret += CountOf(node.Left) + 1;
				}

				prev = node;
				node = node.Parent;
			}

			return ret;
		}
		public Node FindByIndex(int index)
		{
			var current = root_;
			var currentIndex = current.Size - CountOf(current.Right) - 1;
			while (currentIndex != index) {
				if (currentIndex > index) {
					current = current.Left;
					if (current is null) {
						break;
					}

					currentIndex -= CountOf(current.Right) + 1;
				} else {
					current = current.Right;
					if (current is null) {
						break;
					}

					currentIndex += CountOf(current.Left) + 1;
				}
			}

			return current;
		}

		public void Add(T item)
		{
			if (root_ is null) {
				root_ = new Node(item, false);
				return;
			}

			Node current = root_;
			Node parent = null;
			Node grandParent = null;
			Node greatGrandParent = null;
			int order = 0;
			while (current is null == false) {
				order = comparer_(item, current.Item);
				if (order == 0 && isMulti_ == false) {
					root_.IsRed = false;
					return;
				}

				if (Is4Node(current)) {
					Split4Node(current);
					if (IsNonNullRed(parent) == true) {
						InsertionBalance(current, ref parent, grandParent, greatGrandParent);
					}
				}

				greatGrandParent = grandParent;
				grandParent = parent;
				parent = current;
				current = (order < 0) ? current.Left : current.Right;
			}

			Node node = new Node(item, true);
			if (order >= 0) {
				parent.Right = node;
			} else {
				parent.Left = node;
			}

			if (parent.IsRed) {
				InsertionBalance(node, ref parent, grandParent, greatGrandParent);
			}

			root_.IsRed = false;
		}

		public bool RemoveAt(int index) => Remove(FindByIndex(index).Item);
		public bool Remove(T item)
		{
			if (root_ is null) {
				return false;
			}

			Node current = root_;
			Node parent = null;
			Node grandParent = null;
			Node match = null;
			Node parentOfMatch = null;
			bool foundMatch = false;
			while (current is null == false) {
				if (Is2Node(current)) {
					if (parent is null) {
						current.IsRed = true;
					} else {
						Node sibling = GetSibling(current, parent);
						if (sibling.IsRed) {
							if (parent.Right == sibling) {
								RotateLeft(parent);
							} else {
								RotateRight(parent);
							}

							parent.IsRed = true;
							sibling.IsRed = false;
							ReplaceChildOrRoot(grandParent, parent, sibling);
							grandParent = sibling;
							if (parent == match) {
								parentOfMatch = sibling;
							}

							sibling = (parent.Left == current) ? parent.Right : parent.Left;
						}

						if (Is2Node(sibling)) {
							Merge2Nodes(parent);
						} else {
							TreeRotation rotation = GetRotation(parent, current, sibling);
							Node newGrandParent = Rotate(parent, rotation);
							newGrandParent.IsRed = parent.IsRed;
							parent.IsRed = false;
							current.IsRed = true;
							ReplaceChildOrRoot(grandParent, parent, newGrandParent);
							if (parent == match) {
								parentOfMatch = newGrandParent;
							}
						}
					}
				}

				int order = foundMatch ? -1 : comparer_(item, current.Item);
				if (order == 0) {
					foundMatch = true;
					match = current;
					parentOfMatch = parent;
				}

				grandParent = parent;
				parent = current;
				current = order < 0 ? current.Left : current.Right;
			}

			if (match is null == false) {
				ReplaceNode(match, parentOfMatch, parent, grandParent);
			}

			if (root_ is null == false) {
				root_.IsRed = false;
			}

			return foundMatch;
		}

		public void Clear()
		{
			root_ = null;
		}

		public void CopyTo(T[] array, int arrayIndex)
		{
			foreach (var item in this) {
				array[arrayIndex++] = item;
			}
		}

		public (Node node, int index) BinarySearch(T item, bool isLowerBound)
		{
			Node right = null;
			int ri = -1;
			Node current = root_;
			if (current is null) {
				return (right, ri);
			}

			ri = 0;
			while (current is null == false) {
				var order = comparer_(item, current.Item);
				if (order < 0 || (isLowerBound && order == 0)) {
					right = current;
					current = current.Left;
				} else {
					ri += CountOf(current.Left) + 1;
					current = current.Right;
				}
			}

			return (right, ri);
		}
		public Node FindNodeLowerBound(T item) => BinarySearch(item, true).node;
		public Node FindNodeUpperBound(T item) => BinarySearch(item, false).node;
		public T LowerBoundItem(T item) => BinarySearch(item, true).node.Item;
		public T UpperBoundItem(T item) => BinarySearch(item, false).node.Item;
		public int LowerBoundIndex(T item) => BinarySearch(item, true).index;
		public int UpperBoundIndex(T item) => BinarySearch(item, false).index;

		bool ICollection<T>.IsReadOnly => false;
		public IEnumerable<T> Reversed()
		{
			var e = new Enumerator(this, true, null);
			while (e.MoveNext()) {
				yield return e.Current;
			}
		}
		public IEnumerable<T> Enumerate(Node from) => Enumerate(from, false);
		public IEnumerable<T> Enumerate(Node from, bool reverse)
		{
			if (from is null) {
				yield break;
			}

			var e = new Enumerator(this, reverse, from);
			while (e.MoveNext()) {
				yield return e.Current;
			}
		}
		public Enumerator GetEnumerator() => new Enumerator(this);
		IEnumerator<T> IEnumerable<T>.GetEnumerator() => new Enumerator(this);
		System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => new Enumerator(this);

		private static int CountOf(Node node) => node is null ? 0 : node.Size;
		private static bool Is2Node(Node node) => IsNonNullBlack(node) && IsNullOrBlack(node.Left) && IsNullOrBlack(node.Right);
		private static bool Is4Node(Node node) => IsNonNullRed(node.Left) && IsNonNullRed(node.Right);
		private static bool IsNonNullRed(Node node) => node is null == false && node.IsRed;
		private static bool IsNonNullBlack(Node node) => node is null == false && !node.IsRed;
		private static bool IsNullOrBlack(Node node) => node is null || !node.IsRed;

		private static void Merge2Nodes(Node parent)
		{
			parent.IsRed = false;
			parent.Left.IsRed = true;
			parent.Right.IsRed = true;
		}

		private static void Split4Node(Node node)
		{
			node.IsRed = true;
			node.Left.IsRed = false;
			node.Right.IsRed = false;
		}

		private static Node GetSibling(Node node, Node parent)
		{
			return parent.Left == node ? parent.Right : parent.Left;
		}

		private static Node Rotate(Node node, TreeRotation rotation)
		{
			switch (rotation) {
				case TreeRotation.Right:
					node.Left.Left.IsRed = false;
					return RotateRight(node);

				case TreeRotation.Left:
					node.Right.Right.IsRed = false;
					return RotateLeft(node);

				case TreeRotation.RightLeft:
					return RotateRightLeft(node);

				case TreeRotation.LeftRight:
					return RotateLeftRight(node);

				default: throw new InvalidOperationException();
			}
		}

		private static Node RotateLeft(Node node)
		{
			Node child = node.Right;
			node.Right = child.Left;
			child.Left = node;
			return child;
		}

		private static Node RotateLeftRight(Node node)
		{
			Node child = node.Left;
			Node grandChild = child.Right;
			node.Left = grandChild.Right;
			grandChild.Right = node;
			child.Right = grandChild.Left;
			grandChild.Left = child;
			return grandChild;
		}

		private static Node RotateRight(Node node)
		{
			Node child = node.Left;
			node.Left = child.Right;
			child.Right = node;
			return child;
		}

		private static Node RotateRightLeft(Node node)
		{
			Node child = node.Right;
			Node grandChild = child.Left;
			node.Right = grandChild.Left;
			grandChild.Left = node;
			child.Left = grandChild.Right;
			grandChild.Right = child;
			return grandChild;
		}

		private static TreeRotation GetRotation(Node parent, Node current, Node sibling)
		{
			if (IsNonNullRed(sibling.Left)) {
				if (parent.Left == current) {
					return TreeRotation.RightLeft;
				}

				return TreeRotation.Right;
			} else {
				if (parent.Left == current) {
					return TreeRotation.Left;
				}

				return TreeRotation.LeftRight;
			}
		}

		private T[] InitializeArray(IEnumerable<T> collection)
		{
			T[] array;
			if (isMulti_) {
				array = collection.ToArray();
				Array.Sort(array, comparer_);
			} else {
				var list = new List<T>(collection);
				list.Sort(comparer_);
				for (int i = list.Count - 1; i > 0; i--) {
					if (comparer_(list[i - 1], list[i]) == 0) {
						list.RemoveAt(i);
					}
				}

				array = list.ToArray();
			}

			return array;
		}

		private static Node ConstructRootFromSortedArray(T[] arr, int startIndex, int endIndex, Node redNode)
		{
			int size = endIndex - startIndex + 1;
			Node root;
			switch (size) {
				case 0:
					return null;

				case 1:
					root = new Node(arr[startIndex], false);
					if (redNode is null == false) {
						root.Left = redNode;
					}

					break;

				case 2:
					root = new Node(arr[startIndex], false) { Right = new Node(arr[endIndex], true) };
					if (redNode is null == false) {
						root.Left = redNode;
					}

					break;

				case 3:
					root = new Node(arr[startIndex + 1], false) {
						Left = new Node(arr[startIndex], false),
						Right = new Node(arr[endIndex], false)
					};

					if (redNode is null == false) {
						root.Left.Left = redNode;
					}

					break;

				default:
					int midpt = ((startIndex + endIndex) / 2);
					root = new Node(arr[midpt], false) {
						Left = ConstructRootFromSortedArray(arr, startIndex, midpt - 1, redNode),
						Right = size % 2 == 0
							? ConstructRootFromSortedArray(arr, midpt + 2, endIndex, new Node(arr[midpt + 1], true))
							: ConstructRootFromSortedArray(arr, midpt + 1, endIndex, null)
					};

					break;
			}

			return root;
		}

		private void ReplaceNode(Node match, Node parentOfMatch, Node succesor, Node parentOfSuccesor)
		{
			if (succesor == match) {
				succesor = match.Left;
			} else {
				if (succesor.Right is null == false) {
					succesor.Right.IsRed = false;
				}

				if (parentOfSuccesor != match) {
					parentOfSuccesor.Left = succesor.Right;
					succesor.Right = match.Right;
				}

				succesor.Left = match.Left;
			}

			if (succesor is null == false) {
				succesor.IsRed = match.IsRed;
			}

			ReplaceChildOrRoot(parentOfMatch, match, succesor);
		}

		private void InsertionBalance(Node current, ref Node parent, Node grandParent, Node greatGrandParent)
		{
			bool parentIsOnRight = grandParent.Right == parent;
			bool currentIsOnRight = parent.Right == current;
			Node newChildOfGreatGrandParent;
			if (parentIsOnRight == currentIsOnRight) {
				newChildOfGreatGrandParent = currentIsOnRight ? RotateLeft(grandParent) : RotateRight(grandParent);
			} else {
				newChildOfGreatGrandParent = currentIsOnRight ? RotateLeftRight(grandParent) : RotateRightLeft(grandParent);
				parent = greatGrandParent;
			}

			grandParent.IsRed = true;
			newChildOfGreatGrandParent.IsRed = false;
			ReplaceChildOrRoot(greatGrandParent, grandParent, newChildOfGreatGrandParent);
		}

		private void ReplaceChildOrRoot(Node parent, Node child, Node newChild)
		{
			if (parent is null == false) {
				if (parent.Left == child) {
					parent.Left = newChild;
				} else {
					parent.Right = newChild;
				}
			} else {
				root_ = newChild;
			}
		}

		public struct Enumerator : IEnumerator<T>
		{
			private readonly RedBlackTree<T> tree_;
			private readonly Stack<Node> stack_;
			private readonly bool reverse_;
			private Node current_;

			internal Enumerator(RedBlackTree<T> set)
				: this(set, false, null)
			{
			}

			internal Enumerator(RedBlackTree<T> set, bool reverse, Node startNode)
			{
				tree_ = set;
				stack_ = new Stack<Node>(2 * Log2(tree_.Count + 1));
				current_ = null;
				reverse_ = reverse;
				if (startNode is null) {
					IntializeAll();
				} else {
					Intialize(startNode);
				}
			}

			private void IntializeAll()
			{
				var node = tree_.root_;
				while (node is null == false) {
					var next = reverse_ ? node.Right : node.Left;
					stack_.Push(node);
					node = next;
				}
			}

			private void Intialize(Node startNode)
			{
				if (startNode is null) {
					throw new InvalidOperationException(nameof(startNode) + "is null");
				}

				current_ = null;
				var node = startNode;
				var list = new List<Node>(Log2(tree_.Count + 1));
				var comparer = tree_.comparer_;
				if (reverse_) {
					while (node is null == false) {
						list.Add(node);
						var parent = node.Parent;
						if (parent is null || parent.Left == node) {
							node = parent;
							break;
						}

						node = parent;
					}

					while (node is null == false) {
						var parent = node.Parent;
						if (parent is null || parent.Right == node) {
							node = parent;
							break;
						}

						node = parent;
					}

					while (node is null == false) {
						if (comparer(startNode.Item, node.Item) >= 0) {
							list.Add(node);
						}
						node = node.Parent;
					}
				} else {
					while (node is null == false) {
						list.Add(node);
						var parent = node.Parent;
						if (parent is null || parent.Right == node) {
							node = parent;
							break;
						}

						node = parent;
					}

					while (node is null == false) {
						var parent = node.Parent;
						if (parent is null || parent.Left == node) {
							node = parent;
							break;
						}

						node = parent;
					}

					while (node is null == false) {
						if (comparer(startNode.Item, node.Item) <= 0) {
							list.Add(node);
						}
						node = node.Parent;
					}
				}

				list.Reverse();
				foreach (var n in list) {
					stack_.Push(n);
				}
			}

			private static int Log2(int num) => num == 0 ? 0 : BitOperations.Log2((uint)num) + 1;
			object System.Collections.IEnumerator.Current => Current;
			public T Current => current_ is null ? default : current_.Item;
			public bool MoveNext()
			{
				if (stack_.Count == 0) {
					current_ = null;
					return false;
				}

				current_ = stack_.Pop();
				var node = reverse_ ? current_.Left : current_.Right;
				while (node is null == false) {
					var next = reverse_ ? node.Right : node.Left;
					stack_.Push(node);
					node = next;
				}

				return true;
			}

			public void Reset() => throw new NotSupportedException();

			public void Dispose()
			{
			}
		}

		public class Node
		{
			public bool IsRed { get; set; }
			public T Item { get; set; }
			public Node Parent { get; private set; }

			private Node _Left;
			public Node Left
			{
				get { return _Left; }
				set
				{
					_Left = value;
					if (value is null == false) {
						value.Parent = this;
					}
					for (var cur = this; cur is null == false; cur = cur.Parent) {
						if (!cur.UpdateSize()) {
							break;
						}
						if (cur.Parent is null == false && cur.Parent.Left != cur && cur.Parent.Right != cur) {
							cur.Parent = null;
							break;
						}
					}
				}
			}

			private Node _Right;
			public Node Right
			{
				get { return _Right; }
				set
				{
					_Right = value;
					if (value is null == false) {
						value.Parent = this;
					}
					for (var cur = this; cur is null == false; cur = cur.Parent) {
						if (!cur.UpdateSize()) {
							break;
						}
						if (cur.Parent is null == false && cur.Parent.Left != cur && cur.Parent.Right != cur) {
							cur.Parent = null;
							break;
						}
					}
				}
			}
			public int Size { get; private set; } = 1;
			public Node(T item, bool isRed)
			{
				Item = item;
				IsRed = isRed;
			}

			public bool UpdateSize()
			{
				var oldsize = Size;
				var size = 1;
				if (Left is null == false) {
					size += Left.Size;
				}
				if (Right is null == false) {
					size += Right.Size;
				}

				Size = size;
				return oldsize != size;
			}
			public override string ToString() => $"Size = {Size}, Item = {Item}";
		}

		private enum TreeRotation : byte
		{
			Left = 1,
			Right = 2,
			RightLeft = 3,
			LeftRight = 4,
		}
	}

	public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
	{
		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 (ContainsKey(key) == false) {
					base[key] = initialzier_(key);
				}

				return base[key];
			}

			set { base[key] = value; }
		}
	}

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

		public static ModInt Inverse(ModInt value) => Pow(value, P - 2);
		public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);
		public static ModInt Pow(long value, long k)
		{
			long ret = 1;
			for (k %= P - 1; k > 0; k >>= 1, value = value * value % P) {
				if ((k & 1) == 1) {
					ret = ret * value % P;
				}
			}
			return new ModInt(ret);
		}

		private long value_;

		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)
		{
			long exp = P - 2;
			while (exp > 0) {
				if (exp % 2 > 0) {
					lhs *= rhs;
				}

				rhs *= rhs;
				exp /= 2;
			}

			return lhs;
		}

		public static implicit operator ModInt(long n) => new ModInt(n, true);
		public long ToLong() => value_;
		public override string ToString() => value_.ToString();
	}

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

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

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[] Array<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToArray();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static List<T> List<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToList();

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

			return array;
		}
		[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[,,] Array3<T>(int n1, int n2, int n3, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						array[i1, i2, i3] = init;
					}
				}
			}

			return array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T init)
			where T : struct
		{
			var array = new T[n1, n2, n3, n4];
			for (int i1 = 0; i1 < n1; i1++) {
				for (int i2 = 0; i2 < n2; i2++) {
					for (int i3 = 0; i3 < n3; i3++) {
						for (int i4 = 0; i4 < n4; i4++) {
							array[i1, i2, i3, i4] = init;
						}
					}
				}
			}

			return array;
		}

		private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt4(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			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) {
					action(d4i, d4j);
				}
			}
		}

		private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void DoAt8(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			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) {
					action(d8i, d8j);
				}
			}
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static void ForEachSubBit(int bit, Action<int> action)
		{
			for (int sub = bit; sub >= 0; sub--) {
				sub &= bit;
				action(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 Join<T>(this IEnumerable<T> values, string separator = "")
			=> string.Join(separator, values);
	}

	public class Scanner
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private readonly Func<string> reader_;
		private string[] buf_;
		private int index_;

		public Scanner(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string NextLine() => reader_();
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int() => int.Parse(Next());
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public int Int(int offset) => int.Parse(Next()) + 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[] ArrayInt(int length, int offset = 0)
		{
			int[] Array = new int[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Int(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long() => long.Parse(Next());
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public long Long(long offset) => long.Parse(Next()) + 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[] ArrayLong(int length, long offset = 0)
		{
			long[] Array = new long[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Long(offset);
			}
			return Array;
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double() => double.Parse(Next());
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public double Double(double offset) => double.Parse(Next()) + 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[] ArrayDouble(int length, double offset = 0)
		{
			double[] Array = new double[length];
			for (int i = 0; i < length; i++) {
				Array[i] = Double(offset);
			}
			return Array;
		}

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

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