#ifndef __SEGMENT_TREE_H__
#define __SEGMENT_TREE_H__

/* updated: 2020-07-23 */

#include <vector>
#include <functional>
#include <algorithm>
#include <limits>
#include <cassert>
#include <iostream>
#include <string>
#include <sstream>

/*-- てんぷれ --*/
struct HogeMonoid {
	using Type = int;
	static Type identity() {
		return Type();
	}
	static Type operate(const Type& left, const Type& right) {
		return Type();
	}
};

/*--------------*/

namespace Monoid {
	template <class T>
	struct Sum {
		using Type = T;
		static Type identity() {
			return T();
		}
		static Type operate(const Type& left, const Type& right) {
			return left + right;
		}
	};

	template <class T>
	struct Minimum {
		using Type = T;
		static Type identity() {
			return std::numeric_limits<Type>::max();
		}
		static Type operate(const Type& left, const Type& right) {
			return std::min(left, right);
		}
	};

	template <class T>
	struct Maximum {
		using Type = T;
		static Type identity() {
			return std::numeric_limits<Type>::lowest();
		}
		static Type operate(const Type& left, const Type& right) {
			return std::max(left, right);
		}
	};
}

/* 注意: コンストラクタの引数は size でない */
template <class Monoid>
class SegmentTree {
	using T = typename Monoid::Type;

	const int level_m;		// 段数 - 1
	const size_t size_m;	// 1 << level_m
	std::vector<T> arr_m;

	static T identity();
	static T operate(const T& left, const T& right);
	bool isValidIndex(size_t pos) const;
	
	static const int Default = -1;

public:
	explicit SegmentTree(int level);
	explicit SegmentTree(int level, const T& init_val);
	void construct();

	void update(size_t pos, const T& value);
	T query(size_t left, size_t right) const;
	T get(size_t pos) const;

	template <class Function>
	int findBoundary(Function func);

	void debug(int level = Default, int cell_width = 4) const;
};

template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::identity() {
	return Monoid::identity();
}

template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::operate(const T& left, const T& right) {
	return Monoid::operate(left, right);
}

template<class Monoid>
inline bool SegmentTree<Monoid>::isValidIndex(size_t pos) const {
	return pos >= 0 && pos < size_m;
}

/* 1 << level 要素の segment tree_m を作成 */
template<class Monoid>
SegmentTree<Monoid>::SegmentTree(int level)
	: level_m(level), size_m(1 << level) {
	assert(level < 32);
	arr_m.resize(size_m * 2, identity());
}

template<class Monoid>
inline SegmentTree<Monoid>::SegmentTree(int level, const T& init_val)
	: level_m(level), size_m(1 << level) {
	assert(level < 32);

	arr_m.resize(size_m * 2, init_val);
	construct();
}

/*
// 次が成立するように木を構築する, 初期化用 O(N)
// arr[i] = operate(arr[i * 2], arr[i * 2 + 1])
*/
template<class Monoid>
void SegmentTree<Monoid>::construct() {
	for (int i = size_m - 1; i > 0; --i) {
		arr_m[i] = operate(arr_m[i * 2], arr_m[i * 2 + 1]);
	}
}

/* pos 番目の要素を value に書き換える */
template<class Monoid>
void SegmentTree<Monoid>::update(size_t pos, const T& value) {
	pos += size_m;
	arr_m[pos] = value;
	pos >>= 1;
	while (pos > 0) {
		arr_m[pos] = operate(arr_m[pos * 2], arr_m[pos * 2 + 1]);
		pos >>= 1;
	}
}

/* [left, right) に存在する全要素の(指定した演算に関する)積をとる */
template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::query(size_t left, size_t right) const {
	assert(left <= right);

	T tmp_left = identity();
	T tmp_right = identity();
	left += size_m;
	right += size_m;
	while (left < right) {
		if (left & 1) tmp_left = operate(tmp_left, arr_m[left++]);
		if (right & 1) tmp_right = operate(arr_m[--right], tmp_right);
		left >>= 1;
		right >>= 1;
	}

	return operate(tmp_left, tmp_right);
}

/* pos 番目の要素を直接取得 */
template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::get(size_t pos) const {
	return arr_m[size_m + pos];
}

/*
// SegmentTree 上の二分探索 O(log N)
// func(query(0, i - 1)) == true かつ func(query(0, i)) == false となる i を(1つ)返す (0 <= i <= size)
*/
template<class Monoid>
template<class Function>
int SegmentTree<Monoid>::findBoundary(Function func) {
	if (func(arr_m[1])) return size_m;		// 全要素 true のときは先に除外

	T cumul = identity();
	int right = 1;
	while (right < size_m) {
		right <<= 1;
		T cumul_next = operate(cumul, arr_m[right]);
		if (func(cumul_next)) {
			++right;
			cumul = cumul_next;
		}
	}
	return right - size_m;
}

/* デバッグ出力用; level = -1 とすると全要素出力 */
template<class Monoid>
void SegmentTree<Monoid>::debug(int level, int cell_width) const {
	/* 値 value を幅 max_length になるように成形して文字列化 */
	auto to_str_fixed = [](const T& value, int max_length) -> std::string {
		std::ostringstream osst;
		osst << value;

		std::string str = osst.str();
		std::string output;
		if (str.size() > max_length) {
			for (int i = 0; i < max_length; ++i) {
				output += '#';
			}
		} else {
			const int remain_length = max_length - str.size();
			const int right_blank = remain_length / 2;
			const int left_blank = remain_length - right_blank;
			for (int i = 0; i < left_blank; ++i) {
				output += ' ';
			}
			output += str;
			for (int i = 0; i < right_blank; ++i) {
				output += ' ';
			}
		}
		return output;
	};

	if (level == Default) {			// デフォルト値への対処
		level = level_m;
	}

	if (level > level_m) {
		level = level_m;
	}

	std::vector<std::string> output_strs;

	for (int lev = 0; lev <= level; ++lev) {
		std::string output;
		const int front = size_m >> lev;

		output += '|';
		for (int i = 0; i < (1 << (level - lev)); ++i) {
			output += to_str_fixed(arr_m[front + i], cell_width);
			output += '|';
		}
		output_strs.push_back(output);

		cell_width = cell_width * 2 + 1;
	}

	reverse(output_strs.begin(), output_strs.end());
	for (const std::string& output : output_strs) {
		std::cout << output << std::endl;
	}
}

#endif
#ifndef __INTMOD_H__
#define __INTMOD_H__

/* updated: 2020-06-24 */

#include <iostream>
#include <cassert>
#include <vector>
#include <cstdint>
#include <functional>
#include <unordered_map>

template <uint32_t Mod>
class IntMod {
	static_assert(Mod != 0, "Mod must not be 0");
	static_assert(Mod < 0x80000000, "Mod must be less than 0x80000000");

private:
	uint32_t value_m;

public:
	IntMod();
	IntMod(int32_t value);
	IntMod(uint32_t value);
	IntMod(int64_t value);
	IntMod(uint64_t value);

	const IntMod<Mod>& operator+() const;
	IntMod<Mod> operator-() const;
	IntMod<Mod>& operator++();
	IntMod<Mod>& operator--();
	IntMod<Mod> operator++(int dummy);
	IntMod<Mod> operator--(int dummy);
	IntMod<Mod>& operator+=(const IntMod<Mod>& right);
	IntMod<Mod>& operator-=(const IntMod<Mod>& right);
	IntMod<Mod>& operator*=(const IntMod<Mod>& right);
	IntMod<Mod>& operator/=(const IntMod<Mod>& right);
	IntMod<Mod> operator[](uint64_t exp) const;

	static IntMod<Mod> add(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> subtract(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> multiply(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> divide(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static int compare(const IntMod<Mod>& a, const IntMod<Mod>& b);

	IntMod<Mod> inverse() const;
	IntMod<Mod> power(uint64_t exp) const;
	uint32_t get() const;

	static IntMod<Mod> fact(uint32_t num);
	static IntMod<Mod> invFact(uint32_t num);
	static IntMod<Mod> combi(uint32_t n, uint32_t r);
	static IntMod<Mod> homcombi(uint32_t n, uint32_t r);
	static IntMod<Mod> perm(uint32_t n, uint32_t r);
	static std::vector<IntMod<Mod>> calcInverseList(int max);
};

template <uint32_t Mod>
IntMod<Mod> operator+(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator-(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator*(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator/(const IntMod<Mod>& a, const IntMod<Mod>& b);

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const IntMod<Mod>& a, const Int& b);

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const Int& a, const IntMod<Mod>& b);

template <uint32_t Mod>
bool operator==(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator!=(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator<(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator<=(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator>(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator>=(const IntMod<Mod>& a, const IntMod<Mod>& b);

template <uint32_t Mod, class Int>
bool operator==(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
bool operator!=(const IntMod<Mod>& a, const Int& b);

template <uint32_t Mod>
std::istream& operator>>(std::istream& ist, IntMod<Mod>& object);
template <uint32_t Mod>
std::ostream& operator<<(std::ostream& ost, const IntMod<Mod>& object);

using MInt = IntMod<1000000007>;
using MF = IntMod<998244353>;

// for unordered_set/map
namespace std {
	template <uint32_t Mod>
	struct hash<IntMod<Mod>> {
		size_t operator() (const IntMod<Mod>& object) const {
			return std::hash<uint32_t>()(object.get());
		}
	};
}

/*------------------------------------*/
/*-------   Implementations   --------*/
/*------------------------------------*/

template <uint32_t Mod>
IntMod<Mod>::IntMod() 
	: value_m(0) {
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(int32_t value) {
	int32_t tmp = value % int32_t(Mod);
	value_m = tmp >= 0 ? tmp : Mod - uint32_t(-tmp);
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(uint32_t value)
	: value_m(value % Mod) {
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(int64_t value) {
	int32_t tmp = value % int64_t(Mod);
	value_m = tmp >= 0 ? tmp : Mod - uint32_t(-tmp);
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(uint64_t value)
	: value_m(value % Mod) {
}

template <uint32_t Mod>
const IntMod<Mod>& IntMod<Mod>::operator+() const {
	return *this;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator-() const {
	IntMod<Mod> ret;
	if (value_m == 0) {
		ret.value_m = 0;
	} else {
		ret.value_m = Mod - value_m;
	}
	return ret;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator++() {
	++value_m;
	if (value_m == Mod) {
		value_m = 0;
	}
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator--() {
	if (value_m == 0) {
		value_m = Mod;
	}
	--value_m;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator++(int dummy) {
	IntMod ret(*this);
	++*this;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator--(int dummy) {
	IntMod ret(*this);
	--*this;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator+=(const IntMod<Mod>& right) {
	value_m += right.value_m;		// note: value_m < 0x80000000
	if (value_m >= Mod) {
		value_m -= Mod;
	}
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator-=(const IntMod<Mod>& right) {
	if (value_m < right.value_m) {
		value_m += Mod;				// note: value_m < 0x80000000
	}
	value_m -= right.value_m;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator*=(const IntMod<Mod>& right) {
	value_m = (uint64_t(value_m) * right.value_m) % Mod;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator/=(const IntMod<Mod>& right) {
	*this *= right.inverse();
	return *this;
}

// for power
template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator[](uint64_t exp) const {
	return power(exp);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::add(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret += b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::subtract(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret -= b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::multiply(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret *= b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::divide(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret /= b;
	return ret;
}

template <uint32_t Mod>
int IntMod<Mod>::compare(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	if (a.value_m < b.value_m) return -1;
	if (a.value_m > b.value_m) return 1;
	return 0;
}

/* Mod must be a prime. */
template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::inverse() const {
	assert(value_m != 0);
	return power(Mod - 2);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::power(uint64_t exp) const {
	IntMod<Mod> product(1);
	IntMod<Mod> factor(*this);
	while (exp > 0) {
		if (exp & 1) {
			product *= factor;
		}
		factor *= factor;
		exp >>= 1;
	}
	return product;
}

template <uint32_t Mod>
uint32_t IntMod<Mod>::get() const {
	return value_m;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::fact(uint32_t num) {
	assert(num <= 0x80000000);

	static std::vector<IntMod<Mod>> table(1, 1);
	if (table.size() > num) return table[num];

	uint32_t old_size = table.size();
	table.resize(num + 1);
	for (uint32_t i = old_size; i <= num; ++i) {
		table[i] = table[i - 1] * i;
	}
	return table[num];
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::invFact(uint32_t num) {
	assert(num <= 0x80000000);

	static std::vector<IntMod<Mod>> table(1, 1);
	if (table.size() > num) return table[num];

	uint32_t old_size = table.size();
	table.resize(num + 1);
	table[num] = fact(num).inverse();
	for (uint32_t i = num; i >= old_size; --i) {
		table[i - 1] = table[i] * i;
	}
	return table[num];
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::combi(uint32_t n, uint32_t r) {
	assert(n >= r);
	return fact(n) * invFact(n - r) * invFact(r);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::homcombi(uint32_t n, uint32_t r) {
	assert(n != 0 || r == 0);

	if (n == 0 && r == 0) return IntMod<Mod>(1);
	return combi(n + r - 1, r);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::perm(uint32_t n, uint32_t r) {
	assert(n >= r);
	return fact(n) * invFact(n - r);
}

template <uint32_t Mod>
std::vector<IntMod<Mod>> IntMod<Mod>::calcInverseList(int max) {
	assert(max < Mod);

	std::vector<IntMod<Mod>> table(max + 1);
	table[1] = 1;
	for (int i = 2; i <= max; ++i) {
		table[i] = (uint64_t(Mod - Mod / i) * table[Mod % i].get()) % Mod;
	}
	return table;
}

template <uint32_t Mod>
IntMod<Mod> operator+(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::add(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator-(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::subtract(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator*(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::multiply(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator/(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::divide(a, b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const IntMod<Mod>& a, const Int& b) {
	return a + IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const IntMod<Mod>& a, const Int& b) {
	return a - IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const IntMod<Mod>& a, const Int& b) {
	return a * IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const IntMod<Mod>& a, const Int& b) {
	return a / IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) + b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) - b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) * b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) / b;
}

template <uint32_t Mod>
bool operator==(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) == 0;
}

template <uint32_t Mod>
bool operator!=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) != 0;
}

template <uint32_t Mod>
bool operator<(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) < 0;
}

template <uint32_t Mod>
bool operator<=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) <= 0;
}

template <uint32_t Mod>
bool operator>(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) > 0;
}

template <uint32_t Mod>
bool operator>=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) >= 0;
}

template <uint32_t Mod, class Int>
bool operator==(const IntMod<Mod>& a, const Int& b) {
	return a == IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
bool operator!=(const IntMod<Mod>& a, const Int& b) {
	return a != IntMod<Mod>(b);
}

template <uint32_t Mod>
std::istream& operator>>(std::istream& ist, IntMod<Mod>& object) {
	uint64_t value;
	ist >> value;
	object = IntMod<Mod>(value);
	return ist;
}

template <uint32_t Mod>
std::ostream& operator<<(std::ostream& ost, const IntMod<Mod>& object) {
	ost << object.get();
	return ost;
}

#endif

#define _CRT_SECURE_NO_WARNINGS
#define _SCL_SECURE_NO_WARNINGS
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cassert>
#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <utility>
#include <algorithm>
#include <functional>
#include <cmath>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <iomanip>
#include <sstream>
#include <bitset>
#include <limits>
#include <numeric>
#include <valarray>
#include <fstream>
#include <array>
#include <random>
#include <unordered_set>
#include <unordered_map>

using namespace std;
using uint = uint32_t;
using LL = int64_t;
using ULL = uint64_t;
using PP = pair<LL, LL>;
template <typename T> using PriorityQ = priority_queue<T, vector<T>, greater<T> >;
#define REP(i, a, n) for(LL i = (a), i##_max_ = (n); i < i##_max_; ++i)
#define REM(i, a, n) for(LL i = (LL)(n) - 1, i##_min_ = (a); i >= i##_min_; --i)
#define FLOAT fixed << setprecision(16)
#define SPEEDUP { cin.tie(NULL); ios::sync_with_stdio(false); }
const int INF = 0x3FFFFFFF;
const LL INFLL = 0x3FFFFFFF3FFFFFFF;
const double INFD = 1.0e+308;
const double EPS = 1.0e-9;

void YesNo(bool b) { cout << (b ? "Yes" : "No") << endl; }
void YESNO(bool b) { cout << (b ? "YES" : "NO") << endl; }
template <class T, class U> istream& operator>>(istream& ist, pair<T, U>& right) { return ist >> right.first >> right.second; }
template <class T, class U> ostream& operator<<(ostream& ost, const pair<T, U>& right) { return ost << right.first << ' ' << right.second; }
template <class T, class TCompatible, size_t N> void Fill(T(&dest)[N], const TCompatible& val) { fill(dest, dest + N, val); }
template <class T, class TCompatible, size_t M, size_t N> void Fill(T(&dest)[M][N], const TCompatible& val) { for (int i = 0; i < M; ++i) Fill(dest[i], val); }
template <class T> T Next() { T buf; cin >> buf; return buf; }
istream& Ignore(istream& ist) { string s; ist >> s; return ist; }
bool Inside(int i, int j, int h, int w) { return i >= 0 && i < h&& j >= 0 && j < w; }

#ifdef ONLY_MY_ENVIR
#include "Accumulator.h"
#include "Algebraic.h"
#include "BinaryMatrix.h"
#include "BinaryTree.h"
#include "Bipartite.h"
#include "BIT.h"
#include "Compressor.h"
#include "Decompositions.h"
#include "DiscreteLog.h"
#include "DynamicMod.h"
#include "Exponential.h"
#include "Factorization.h"
#include "FFT.h"
#include "FlowSolver.h"
#include "Geometric2D.h"
#include "Geometric2DFloat.h"
#include "Geometric3D.h"
#include "Geometric3DFloat.h"
#include "Graph.h"
#include "GraphUtil.h"
#include "Interpolation.h"
#include "IntMod.h"
#include "KDTree.h"
#include "LazySegmentTree.h"
#include "LIS.h"
#include "List.h"
#include "Math.h"
#include "MathUtil.h"
#include "Matrix.h"
#include "MinCostFlowSolver.h"
#include "MinMax.h"
#include "Numbers.h"
#include "Optimize.h"
#include "Permutation.h"
#include "Polynomial.h"
#include "Position.h"
#include "Random.h"
#include "Range.h"
#include "Rational.h"
#include "RollingHash.h"
#include "RuntimeMod.h"
#include "SegmentTree.h"
#include "SegmentTree2D.h"
#include "Sets.h"
#include "Shortest.h"
#include "SlidingWindow.h"
#include "SpanningTree.h"
#include "StringSearching.h"
#include "SuffixArray.h"
#include "SwitchList.h"
#include "Timer.h"
#include "Tree.h"
#include "TreeUtil.h"
#include "UnionFind.h"
#include "Util.h"
#include "VectorUtil.h"
#endif

#ifdef __GNUC__
typedef __int128 LLL;
istream& operator>>(istream& ist, __int128& val) { LL tmp; ist >> tmp; val = tmp; return ist; }
ostream& operator<<(ostream& ost, __int128 val) { LL tmp = val; ost << tmp; return ost; }
#endif

#include <cctype>
#include <bitset>
#include <iostream>
#include <string>
#include <vector>
using namespace std;

int n, k;
int a[200000];
int main() {
	cin >> n >> k;
	REP(i, 0, n) {
		cin >> a[i];
		--a[i];
	}


	MF base = 1;
	SegmentTree<Monoid::Sum<MF>> seg(18);
	SegmentTree<Monoid::Sum<MF>> cnt(18);
	MF sum = MF(k) * (k - 1) / 2 / 2;
	REP(i, 0, k) {
		seg.update(a[i], base);
		cnt.update(a[i], 1);
	}

	REP(i, k, n) {
		MF res = seg.query(0, a[i]);
		MF tmp = res / k / base;
		MF ttl = cnt.query(a[i], n);
		sum += tmp * (k - 1) / 2;
		sum += ttl - (1 - tmp) * (k - 1) / 2;
		base *= MF(k) / (k - 1);
		seg.update(a[i], base);
		cnt.update(a[i], 1);
	}

	cout << sum << endl;
	return 0;
}

### Prompt

Construct a cpp code solution to the problem outlined:

Given are a permutation p_1, p_2, \dots, p_N of (1, 2, ..., N) and an integer K. Maroon performs the following operation for i = 1, 2, \dots, N - K + 1 in this order:

* Shuffle p_i, p_{i + 1}, \dots, p_{i + K - 1} uniformly randomly.



Find the expected value of the inversion number of the sequence after all the operations are performed, and print it modulo 998244353.

More specifically, from the constraints of this problem, it can be proved that the expected value is always a rational number, which can be represented as an irreducible fraction \frac{P}{Q}, and that the integer R that satisfies R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353 is uniquely determined. Print this R.

Here, the inversion number of a sequence a_1, a_2, \dots, a_N is defined to be the number of ordered pairs (i, j) that satisfy i < j, a_i > a_j.

Constraints

* 2 \leq N \leq 200,000
* 2 \leq K \leq N
* (p_1, p_2, \dots, p_N) is a permutation of (1, 2, \dots, N).
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N K
p_1 p_2 ... p_N


Output

Print the expected value modulo 998244353.

Examples

Input

3 2
1 2 3


Output

1


Input

10 3
1 8 4 9 2 3 7 10 5 6


Output

164091855

### Response

```cpp
#ifndef __SEGMENT_TREE_H__
#define __SEGMENT_TREE_H__

/* updated: 2020-07-23 */

#include <vector>
#include <functional>
#include <algorithm>
#include <limits>
#include <cassert>
#include <iostream>
#include <string>
#include <sstream>

/*-- てんぷれ --*/
struct HogeMonoid {
	using Type = int;
	static Type identity() {
		return Type();
	}
	static Type operate(const Type& left, const Type& right) {
		return Type();
	}
};

/*--------------*/

namespace Monoid {
	template <class T>
	struct Sum {
		using Type = T;
		static Type identity() {
			return T();
		}
		static Type operate(const Type& left, const Type& right) {
			return left + right;
		}
	};

	template <class T>
	struct Minimum {
		using Type = T;
		static Type identity() {
			return std::numeric_limits<Type>::max();
		}
		static Type operate(const Type& left, const Type& right) {
			return std::min(left, right);
		}
	};

	template <class T>
	struct Maximum {
		using Type = T;
		static Type identity() {
			return std::numeric_limits<Type>::lowest();
		}
		static Type operate(const Type& left, const Type& right) {
			return std::max(left, right);
		}
	};
}

/* 注意: コンストラクタの引数は size でない */
template <class Monoid>
class SegmentTree {
	using T = typename Monoid::Type;

	const int level_m;		// 段数 - 1
	const size_t size_m;	// 1 << level_m
	std::vector<T> arr_m;

	static T identity();
	static T operate(const T& left, const T& right);
	bool isValidIndex(size_t pos) const;
	
	static const int Default = -1;

public:
	explicit SegmentTree(int level);
	explicit SegmentTree(int level, const T& init_val);
	void construct();

	void update(size_t pos, const T& value);
	T query(size_t left, size_t right) const;
	T get(size_t pos) const;

	template <class Function>
	int findBoundary(Function func);

	void debug(int level = Default, int cell_width = 4) const;
};

template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::identity() {
	return Monoid::identity();
}

template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::operate(const T& left, const T& right) {
	return Monoid::operate(left, right);
}

template<class Monoid>
inline bool SegmentTree<Monoid>::isValidIndex(size_t pos) const {
	return pos >= 0 && pos < size_m;
}

/* 1 << level 要素の segment tree_m を作成 */
template<class Monoid>
SegmentTree<Monoid>::SegmentTree(int level)
	: level_m(level), size_m(1 << level) {
	assert(level < 32);
	arr_m.resize(size_m * 2, identity());
}

template<class Monoid>
inline SegmentTree<Monoid>::SegmentTree(int level, const T& init_val)
	: level_m(level), size_m(1 << level) {
	assert(level < 32);

	arr_m.resize(size_m * 2, init_val);
	construct();
}

/*
// 次が成立するように木を構築する, 初期化用 O(N)
// arr[i] = operate(arr[i * 2], arr[i * 2 + 1])
*/
template<class Monoid>
void SegmentTree<Monoid>::construct() {
	for (int i = size_m - 1; i > 0; --i) {
		arr_m[i] = operate(arr_m[i * 2], arr_m[i * 2 + 1]);
	}
}

/* pos 番目の要素を value に書き換える */
template<class Monoid>
void SegmentTree<Monoid>::update(size_t pos, const T& value) {
	pos += size_m;
	arr_m[pos] = value;
	pos >>= 1;
	while (pos > 0) {
		arr_m[pos] = operate(arr_m[pos * 2], arr_m[pos * 2 + 1]);
		pos >>= 1;
	}
}

/* [left, right) に存在する全要素の(指定した演算に関する)積をとる */
template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::query(size_t left, size_t right) const {
	assert(left <= right);

	T tmp_left = identity();
	T tmp_right = identity();
	left += size_m;
	right += size_m;
	while (left < right) {
		if (left & 1) tmp_left = operate(tmp_left, arr_m[left++]);
		if (right & 1) tmp_right = operate(arr_m[--right], tmp_right);
		left >>= 1;
		right >>= 1;
	}

	return operate(tmp_left, tmp_right);
}

/* pos 番目の要素を直接取得 */
template<class Monoid>
typename SegmentTree<Monoid>::T SegmentTree<Monoid>::get(size_t pos) const {
	return arr_m[size_m + pos];
}

/*
// SegmentTree 上の二分探索 O(log N)
// func(query(0, i - 1)) == true かつ func(query(0, i)) == false となる i を(1つ)返す (0 <= i <= size)
*/
template<class Monoid>
template<class Function>
int SegmentTree<Monoid>::findBoundary(Function func) {
	if (func(arr_m[1])) return size_m;		// 全要素 true のときは先に除外

	T cumul = identity();
	int right = 1;
	while (right < size_m) {
		right <<= 1;
		T cumul_next = operate(cumul, arr_m[right]);
		if (func(cumul_next)) {
			++right;
			cumul = cumul_next;
		}
	}
	return right - size_m;
}

/* デバッグ出力用; level = -1 とすると全要素出力 */
template<class Monoid>
void SegmentTree<Monoid>::debug(int level, int cell_width) const {
	/* 値 value を幅 max_length になるように成形して文字列化 */
	auto to_str_fixed = [](const T& value, int max_length) -> std::string {
		std::ostringstream osst;
		osst << value;

		std::string str = osst.str();
		std::string output;
		if (str.size() > max_length) {
			for (int i = 0; i < max_length; ++i) {
				output += '#';
			}
		} else {
			const int remain_length = max_length - str.size();
			const int right_blank = remain_length / 2;
			const int left_blank = remain_length - right_blank;
			for (int i = 0; i < left_blank; ++i) {
				output += ' ';
			}
			output += str;
			for (int i = 0; i < right_blank; ++i) {
				output += ' ';
			}
		}
		return output;
	};

	if (level == Default) {			// デフォルト値への対処
		level = level_m;
	}

	if (level > level_m) {
		level = level_m;
	}

	std::vector<std::string> output_strs;

	for (int lev = 0; lev <= level; ++lev) {
		std::string output;
		const int front = size_m >> lev;

		output += '|';
		for (int i = 0; i < (1 << (level - lev)); ++i) {
			output += to_str_fixed(arr_m[front + i], cell_width);
			output += '|';
		}
		output_strs.push_back(output);

		cell_width = cell_width * 2 + 1;
	}

	reverse(output_strs.begin(), output_strs.end());
	for (const std::string& output : output_strs) {
		std::cout << output << std::endl;
	}
}

#endif
#ifndef __INTMOD_H__
#define __INTMOD_H__

/* updated: 2020-06-24 */

#include <iostream>
#include <cassert>
#include <vector>
#include <cstdint>
#include <functional>
#include <unordered_map>

template <uint32_t Mod>
class IntMod {
	static_assert(Mod != 0, "Mod must not be 0");
	static_assert(Mod < 0x80000000, "Mod must be less than 0x80000000");

private:
	uint32_t value_m;

public:
	IntMod();
	IntMod(int32_t value);
	IntMod(uint32_t value);
	IntMod(int64_t value);
	IntMod(uint64_t value);

	const IntMod<Mod>& operator+() const;
	IntMod<Mod> operator-() const;
	IntMod<Mod>& operator++();
	IntMod<Mod>& operator--();
	IntMod<Mod> operator++(int dummy);
	IntMod<Mod> operator--(int dummy);
	IntMod<Mod>& operator+=(const IntMod<Mod>& right);
	IntMod<Mod>& operator-=(const IntMod<Mod>& right);
	IntMod<Mod>& operator*=(const IntMod<Mod>& right);
	IntMod<Mod>& operator/=(const IntMod<Mod>& right);
	IntMod<Mod> operator[](uint64_t exp) const;

	static IntMod<Mod> add(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> subtract(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> multiply(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static IntMod<Mod> divide(const IntMod<Mod>& a, const IntMod<Mod>& b);
	static int compare(const IntMod<Mod>& a, const IntMod<Mod>& b);

	IntMod<Mod> inverse() const;
	IntMod<Mod> power(uint64_t exp) const;
	uint32_t get() const;

	static IntMod<Mod> fact(uint32_t num);
	static IntMod<Mod> invFact(uint32_t num);
	static IntMod<Mod> combi(uint32_t n, uint32_t r);
	static IntMod<Mod> homcombi(uint32_t n, uint32_t r);
	static IntMod<Mod> perm(uint32_t n, uint32_t r);
	static std::vector<IntMod<Mod>> calcInverseList(int max);
};

template <uint32_t Mod>
IntMod<Mod> operator+(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator-(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator*(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
IntMod<Mod> operator/(const IntMod<Mod>& a, const IntMod<Mod>& b);

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const IntMod<Mod>& a, const Int& b);

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const Int& a, const IntMod<Mod>& b);
template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const Int& a, const IntMod<Mod>& b);

template <uint32_t Mod>
bool operator==(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator!=(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator<(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator<=(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator>(const IntMod<Mod>& a, const IntMod<Mod>& b);
template <uint32_t Mod>
bool operator>=(const IntMod<Mod>& a, const IntMod<Mod>& b);

template <uint32_t Mod, class Int>
bool operator==(const IntMod<Mod>& a, const Int& b);
template <uint32_t Mod, class Int>
bool operator!=(const IntMod<Mod>& a, const Int& b);

template <uint32_t Mod>
std::istream& operator>>(std::istream& ist, IntMod<Mod>& object);
template <uint32_t Mod>
std::ostream& operator<<(std::ostream& ost, const IntMod<Mod>& object);

using MInt = IntMod<1000000007>;
using MF = IntMod<998244353>;

// for unordered_set/map
namespace std {
	template <uint32_t Mod>
	struct hash<IntMod<Mod>> {
		size_t operator() (const IntMod<Mod>& object) const {
			return std::hash<uint32_t>()(object.get());
		}
	};
}

/*------------------------------------*/
/*-------   Implementations   --------*/
/*------------------------------------*/

template <uint32_t Mod>
IntMod<Mod>::IntMod() 
	: value_m(0) {
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(int32_t value) {
	int32_t tmp = value % int32_t(Mod);
	value_m = tmp >= 0 ? tmp : Mod - uint32_t(-tmp);
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(uint32_t value)
	: value_m(value % Mod) {
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(int64_t value) {
	int32_t tmp = value % int64_t(Mod);
	value_m = tmp >= 0 ? tmp : Mod - uint32_t(-tmp);
}

template <uint32_t Mod>
IntMod<Mod>::IntMod(uint64_t value)
	: value_m(value % Mod) {
}

template <uint32_t Mod>
const IntMod<Mod>& IntMod<Mod>::operator+() const {
	return *this;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator-() const {
	IntMod<Mod> ret;
	if (value_m == 0) {
		ret.value_m = 0;
	} else {
		ret.value_m = Mod - value_m;
	}
	return ret;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator++() {
	++value_m;
	if (value_m == Mod) {
		value_m = 0;
	}
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator--() {
	if (value_m == 0) {
		value_m = Mod;
	}
	--value_m;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator++(int dummy) {
	IntMod ret(*this);
	++*this;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator--(int dummy) {
	IntMod ret(*this);
	--*this;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator+=(const IntMod<Mod>& right) {
	value_m += right.value_m;		// note: value_m < 0x80000000
	if (value_m >= Mod) {
		value_m -= Mod;
	}
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator-=(const IntMod<Mod>& right) {
	if (value_m < right.value_m) {
		value_m += Mod;				// note: value_m < 0x80000000
	}
	value_m -= right.value_m;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator*=(const IntMod<Mod>& right) {
	value_m = (uint64_t(value_m) * right.value_m) % Mod;
	return *this;
}

template <uint32_t Mod>
IntMod<Mod>& IntMod<Mod>::operator/=(const IntMod<Mod>& right) {
	*this *= right.inverse();
	return *this;
}

// for power
template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::operator[](uint64_t exp) const {
	return power(exp);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::add(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret += b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::subtract(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret -= b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::multiply(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret *= b;
	return ret;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::divide(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	IntMod<Mod> ret(a);
	ret /= b;
	return ret;
}

template <uint32_t Mod>
int IntMod<Mod>::compare(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	if (a.value_m < b.value_m) return -1;
	if (a.value_m > b.value_m) return 1;
	return 0;
}

/* Mod must be a prime. */
template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::inverse() const {
	assert(value_m != 0);
	return power(Mod - 2);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::power(uint64_t exp) const {
	IntMod<Mod> product(1);
	IntMod<Mod> factor(*this);
	while (exp > 0) {
		if (exp & 1) {
			product *= factor;
		}
		factor *= factor;
		exp >>= 1;
	}
	return product;
}

template <uint32_t Mod>
uint32_t IntMod<Mod>::get() const {
	return value_m;
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::fact(uint32_t num) {
	assert(num <= 0x80000000);

	static std::vector<IntMod<Mod>> table(1, 1);
	if (table.size() > num) return table[num];

	uint32_t old_size = table.size();
	table.resize(num + 1);
	for (uint32_t i = old_size; i <= num; ++i) {
		table[i] = table[i - 1] * i;
	}
	return table[num];
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::invFact(uint32_t num) {
	assert(num <= 0x80000000);

	static std::vector<IntMod<Mod>> table(1, 1);
	if (table.size() > num) return table[num];

	uint32_t old_size = table.size();
	table.resize(num + 1);
	table[num] = fact(num).inverse();
	for (uint32_t i = num; i >= old_size; --i) {
		table[i - 1] = table[i] * i;
	}
	return table[num];
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::combi(uint32_t n, uint32_t r) {
	assert(n >= r);
	return fact(n) * invFact(n - r) * invFact(r);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::homcombi(uint32_t n, uint32_t r) {
	assert(n != 0 || r == 0);

	if (n == 0 && r == 0) return IntMod<Mod>(1);
	return combi(n + r - 1, r);
}

template <uint32_t Mod>
IntMod<Mod> IntMod<Mod>::perm(uint32_t n, uint32_t r) {
	assert(n >= r);
	return fact(n) * invFact(n - r);
}

template <uint32_t Mod>
std::vector<IntMod<Mod>> IntMod<Mod>::calcInverseList(int max) {
	assert(max < Mod);

	std::vector<IntMod<Mod>> table(max + 1);
	table[1] = 1;
	for (int i = 2; i <= max; ++i) {
		table[i] = (uint64_t(Mod - Mod / i) * table[Mod % i].get()) % Mod;
	}
	return table;
}

template <uint32_t Mod>
IntMod<Mod> operator+(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::add(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator-(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::subtract(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator*(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::multiply(a, b);
}

template <uint32_t Mod>
IntMod<Mod> operator/(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::divide(a, b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const IntMod<Mod>& a, const Int& b) {
	return a + IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const IntMod<Mod>& a, const Int& b) {
	return a - IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const IntMod<Mod>& a, const Int& b) {
	return a * IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const IntMod<Mod>& a, const Int& b) {
	return a / IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator+(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) + b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator-(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) - b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator*(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) * b;
}

template <uint32_t Mod, class Int>
IntMod<Mod> operator/(const Int& a, const IntMod<Mod>& b) {
	return IntMod<Mod>(a) / b;
}

template <uint32_t Mod>
bool operator==(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) == 0;
}

template <uint32_t Mod>
bool operator!=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) != 0;
}

template <uint32_t Mod>
bool operator<(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) < 0;
}

template <uint32_t Mod>
bool operator<=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) <= 0;
}

template <uint32_t Mod>
bool operator>(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) > 0;
}

template <uint32_t Mod>
bool operator>=(const IntMod<Mod>& a, const IntMod<Mod>& b) {
	return IntMod<Mod>::compare(a, b) >= 0;
}

template <uint32_t Mod, class Int>
bool operator==(const IntMod<Mod>& a, const Int& b) {
	return a == IntMod<Mod>(b);
}

template <uint32_t Mod, class Int>
bool operator!=(const IntMod<Mod>& a, const Int& b) {
	return a != IntMod<Mod>(b);
}

template <uint32_t Mod>
std::istream& operator>>(std::istream& ist, IntMod<Mod>& object) {
	uint64_t value;
	ist >> value;
	object = IntMod<Mod>(value);
	return ist;
}

template <uint32_t Mod>
std::ostream& operator<<(std::ostream& ost, const IntMod<Mod>& object) {
	ost << object.get();
	return ost;
}

#endif

#define _CRT_SECURE_NO_WARNINGS
#define _SCL_SECURE_NO_WARNINGS
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cassert>
#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <utility>
#include <algorithm>
#include <functional>
#include <cmath>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <iomanip>
#include <sstream>
#include <bitset>
#include <limits>
#include <numeric>
#include <valarray>
#include <fstream>
#include <array>
#include <random>
#include <unordered_set>
#include <unordered_map>

using namespace std;
using uint = uint32_t;
using LL = int64_t;
using ULL = uint64_t;
using PP = pair<LL, LL>;
template <typename T> using PriorityQ = priority_queue<T, vector<T>, greater<T> >;
#define REP(i, a, n) for(LL i = (a), i##_max_ = (n); i < i##_max_; ++i)
#define REM(i, a, n) for(LL i = (LL)(n) - 1, i##_min_ = (a); i >= i##_min_; --i)
#define FLOAT fixed << setprecision(16)
#define SPEEDUP { cin.tie(NULL); ios::sync_with_stdio(false); }
const int INF = 0x3FFFFFFF;
const LL INFLL = 0x3FFFFFFF3FFFFFFF;
const double INFD = 1.0e+308;
const double EPS = 1.0e-9;

void YesNo(bool b) { cout << (b ? "Yes" : "No") << endl; }
void YESNO(bool b) { cout << (b ? "YES" : "NO") << endl; }
template <class T, class U> istream& operator>>(istream& ist, pair<T, U>& right) { return ist >> right.first >> right.second; }
template <class T, class U> ostream& operator<<(ostream& ost, const pair<T, U>& right) { return ost << right.first << ' ' << right.second; }
template <class T, class TCompatible, size_t N> void Fill(T(&dest)[N], const TCompatible& val) { fill(dest, dest + N, val); }
template <class T, class TCompatible, size_t M, size_t N> void Fill(T(&dest)[M][N], const TCompatible& val) { for (int i = 0; i < M; ++i) Fill(dest[i], val); }
template <class T> T Next() { T buf; cin >> buf; return buf; }
istream& Ignore(istream& ist) { string s; ist >> s; return ist; }
bool Inside(int i, int j, int h, int w) { return i >= 0 && i < h&& j >= 0 && j < w; }

#ifdef ONLY_MY_ENVIR
#include "Accumulator.h"
#include "Algebraic.h"
#include "BinaryMatrix.h"
#include "BinaryTree.h"
#include "Bipartite.h"
#include "BIT.h"
#include "Compressor.h"
#include "Decompositions.h"
#include "DiscreteLog.h"
#include "DynamicMod.h"
#include "Exponential.h"
#include "Factorization.h"
#include "FFT.h"
#include "FlowSolver.h"
#include "Geometric2D.h"
#include "Geometric2DFloat.h"
#include "Geometric3D.h"
#include "Geometric3DFloat.h"
#include "Graph.h"
#include "GraphUtil.h"
#include "Interpolation.h"
#include "IntMod.h"
#include "KDTree.h"
#include "LazySegmentTree.h"
#include "LIS.h"
#include "List.h"
#include "Math.h"
#include "MathUtil.h"
#include "Matrix.h"
#include "MinCostFlowSolver.h"
#include "MinMax.h"
#include "Numbers.h"
#include "Optimize.h"
#include "Permutation.h"
#include "Polynomial.h"
#include "Position.h"
#include "Random.h"
#include "Range.h"
#include "Rational.h"
#include "RollingHash.h"
#include "RuntimeMod.h"
#include "SegmentTree.h"
#include "SegmentTree2D.h"
#include "Sets.h"
#include "Shortest.h"
#include "SlidingWindow.h"
#include "SpanningTree.h"
#include "StringSearching.h"
#include "SuffixArray.h"
#include "SwitchList.h"
#include "Timer.h"
#include "Tree.h"
#include "TreeUtil.h"
#include "UnionFind.h"
#include "Util.h"
#include "VectorUtil.h"
#endif

#ifdef __GNUC__
typedef __int128 LLL;
istream& operator>>(istream& ist, __int128& val) { LL tmp; ist >> tmp; val = tmp; return ist; }
ostream& operator<<(ostream& ost, __int128 val) { LL tmp = val; ost << tmp; return ost; }
#endif

#include <cctype>
#include <bitset>
#include <iostream>
#include <string>
#include <vector>
using namespace std;

int n, k;
int a[200000];
int main() {
	cin >> n >> k;
	REP(i, 0, n) {
		cin >> a[i];
		--a[i];
	}


	MF base = 1;
	SegmentTree<Monoid::Sum<MF>> seg(18);
	SegmentTree<Monoid::Sum<MF>> cnt(18);
	MF sum = MF(k) * (k - 1) / 2 / 2;
	REP(i, 0, k) {
		seg.update(a[i], base);
		cnt.update(a[i], 1);
	}

	REP(i, k, n) {
		MF res = seg.query(0, a[i]);
		MF tmp = res / k / base;
		MF ttl = cnt.query(a[i], n);
		sum += tmp * (k - 1) / 2;
		sum += ttl - (1 - tmp) * (k - 1) / 2;
		base *= MF(k) / (k - 1);
		seg.update(a[i], base);
		cnt.update(a[i], 1);
	}

	cout << sum << endl;
	return 0;
}

```