// Begin include: "../../template/template.hpp"
using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// Begin include: "util.hpp"
namespace yamada {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using lld = long double;

template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
template <typename T>
using VVV = vector<vector<vector<T>>>;
template <typename T>
using VVVV = vector<vector<vector<vector<T>>>>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvl = vector<vector<long long>>;
using vvvl = vector<vector<vector<long long>>>;
using vvvvl = vector<vector<vector<vector<long long>>>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxpq = priority_queue<T, vector<T>, less<T>>;

template <typename T, typename U>
struct P : pair<T, U> {
	template <typename... Args>
	P(Args... args) : pair<T, U>(args...) {}

	using pair<T, U>::first;
	using pair<T, U>::second;

	P &operator+=(const P &r) {
		first += r.first;
		second += r.second;
		return *this;
	}
	P &operator-=(const P &r) {
		first -= r.first;
		second -= r.second;
		return *this;
	}
	P &operator*=(const P &r) {
		first *= r.first;
		second *= r.second;
		return *this;
	}
	template <typename S>
	P &operator*=(const S &r) {
		first *= r, second *= r;
		return *this;
	}
	P operator+(const P &r) const { return P(*this) += r; }
	P operator-(const P &r) const { return P(*this) -= r; }
	P operator*(const P &r) const { return P(*this) *= r; }
	template <typename S>
	P operator*(const S &r) const {
		return P(*this) *= r;
	}
	P operator-() const { return P{-first, -second}; }
};

using pl = P<ll, ll>;
using vp = V<pl>;
using vvp = VV<pl>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T, typename U>
inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template <typename T, typename U>
inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template <typename T>
inline T Max(const vector<T> &v) { return *max_element(begin(v), end(v)); }
template <typename T>
inline T Min(const vector<T> &v) { return *min_element(begin(v), end(v)); }
template <typename T>
inline long long Sum(const vector<T> &v) { return accumulate(begin(v), end(v), T(0)); }

template <typename T>
int lb(const vector<T> &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); }
template <typename T>
int ub(const vector<T> &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); }

constexpr long long TEN(int n) {
	long long ret = 1, x = 10;
	for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
	return ret;
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
	vector<T> ret(v.size() + 1);
	if (rev) {
		for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
	} else {
		for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
	}
	return ret;
};

template <typename T>
vector<T> mkuni(const vector<T> &v) {
	vector<T> ret(v);
	sort(ret.begin(), ret.end());
	ret.erase(unique(ret.begin(), ret.end()), ret.end());
	return ret;
}

template <typename F>
vector<int> mkord(int N, F f) {
	vector<int> ord(N);
	iota(begin(ord), end(ord), 0);
	sort(begin(ord), end(ord), f);
	return ord;
}

template <typename T>
vector<int> mkinv(vector<T> &v) {
	int max_val = *max_element(begin(v), end(v));
	vector<int> inv(max_val + 1, -1);
	for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
	return inv;
}

vector<int> mkiota(int n) {
	vector<int> ret(n);
	iota(begin(ret), end(ret), 0);
	return ret;
}

template <typename T>
T mkrev(const T &v) {
	T w{v};
	reverse(begin(w), end(w));
	return w;
}

template <typename T>
bool nxp(T &v) { return next_permutation(begin(v), end(v)); }

// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
	vector<vector<T>> ret;
	vector<T> v;
	auto dfs = [&](auto rc, int i) -> void {
		if (i == (int)a.size()) {
			ret.push_back(v);
			return;
		}
		for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
	};
	dfs(dfs, 0);
	return ret;
}

template <typename T, typename U>
vector<U> Digit(T a, const U &x, int siz = -1) {
	vector<U> ret;
	while (a > 0) {
		ret.emplace_back(a % x);
		a /= x;
	}
	if (siz >= 0) while ((int)ret.size() < siz) ret.emplace_back(0);
	return ret;
}

// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
	T res = I;
	for (; n; f(a = a * a), n >>= 1) {
		if (n & 1) f(res = res * a);
	}
	return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I = T{1}) {
	return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}

template <typename T>
T Rev(const T &v) {
	T res = v;
	reverse(begin(res), end(res));
	return res;
}

template <typename T>
vector<T> Transpose(const vector<T> &v) {
	using U = typename T::value_type;
	if(v.empty()) return {};
	int H = v.size(), W = v[0].size();
	vector res(W, T(H, U{}));
	for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) res[j][i] = v[i][j];
	return res;
}

template <typename T>
vector<T> Rotate(const vector<T> &v, int clockwise = true) {
	using U = typename T::value_type;
	int H = v.size(), W = v[0].size();
	vector res(W, T(H, U{}));
	for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) {
		if (clockwise) res[W - 1 - j][i] = v[i][j];
		else res[j][H - 1 - i] = v[i][j];
	}
	return res;
}

template <typename T, typename F>
T bisect(T ok, T bad, F pred) {
	if (ok == bad) return ok;
	while (bad - ok > 1) { T mid = ok + (bad - ok) / 2; (pred(mid) ? ok : bad) = mid; } 
	return bad;
}

template <typename T, typename F>
T bisect_double(T ok, T bad, F pred, int iter = 100) {
	if (ok == bad) return ok;
	while (iter--) { T mid = ok + (bad - ok) / 2; (pred(mid) ? ok : bad) = mid; } 
	return bad;
}

template <typename T>
bool inLR(T L, T x, T R){ return (L <= x && x < R); }
bool YESNO(bool b) { std::cout << (b ? "YES\n" : "NO\n"); return b; }
bool YesNo(bool b) { std::cout << (b ? "Yes\n" : "No\n"); return b; }

template <typename mint>
std::string toFraction(mint a, int M) {
	for (int deno = 1; deno <= M; deno++) {
		mint inv = ((mint)deno).inverse();
		for (int nume = -M; nume <= M; nume++) {
			mint val = inv * nume;
			if (val == a) {
				if (deno == 1) return std::to_string(nume);
				return std::to_string(nume) + "/" + std::to_string(deno);
			}
		}
	}
	return "NF";
}

template <typename mint>
void mout(mint a, int M = 100) { std::cout << toFraction(a, M) << "\n"; }
template <typename mint>
void mout(std::vector<mint> A, int M = 100) {
	for (int i = 0; i < (int)A.size(); i++) {
		std::cout << toFraction(A[i], M) << (i == (int)A.size() - 1 ? "\n" : " ");
	}
}

bool is_square(uint64_t n) {
	if (n < 2) return true;
	uint64_t r = static_cast<uint64_t>(sqrtl(static_cast<long double>(n)));
	if (r * r == n) return true;
	++r;
	return r * r == n;
}

template <typename T>
struct CumulativeSum {
	std::vector<T> S;
	CumulativeSum(std::vector<T> &A) {
		int N = A.size();
		S.resize(N + 1);
		for (int i = 0; i < N; i++) S[i + 1] = S[i] + A[i];
	}
	T query(int l, int r) { return (l <= r ? S[r] - S[l] : (T)0); }
	inline T operator()(int l, int r) { return query(l, r); }
};

long long floor(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b - (a % b < 0);
}
long long under(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b - (a % b <= 0);
}
long long ceil(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b + (a % b > 0);
}
long long over(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b + (a % b >= 0);
}
long long modulo(long long a, long long b) {
	assert(b > 0);
	long long c = a % b;
	return c < 0 ? c + b : c;
}

} // namespace yamada

// End include: "util.hpp"
// Begin include: "bitop.hpp"
namespace yamada {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
  return __builtin_popcountll(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
  if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace yamada
// End include: "bitop.hpp"
// Begin include: "inout.hpp"
namespace yamada {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
	os << p.first << " " << p.second;
	return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
	is >> p.first >> p.second;
	return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
	int s = (int)v.size();
	for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
	return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
	for (auto &x : v) is >> x;
	return is;
}

istream &operator>>(istream &is, __int128_t &x) {
	string S;
	is >> S;
	x = 0;
	int flag = 0;
	for (auto &c : S) {
		if (c == '-') {
			flag = true;
			continue;
		}
		x *= 10;
		x += c - '0';
	}
	if (flag) x = -x;
	return is;
}

istream &operator>>(istream &is, __uint128_t &x) {
	string S;
	is >> S;
	x = 0;
	for (auto &c : S) {
		x *= 10;
		x += c - '0';
	}
	return is;
}

ostream &operator<<(ostream &os, __int128_t x) {
	if (x == 0) return os << 0;
	if (x < 0) os << '-', x = -x;
	string S;
	while (x) S.push_back('0' + x % 10), x /= 10;
	reverse(begin(S), end(S));
	return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
	if (x == 0) return os << 0;
	string S;
	while (x) S.push_back('0' + x % 10), x /= 10;
	reverse(begin(S), end(S));
	return os << S;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
	cin >> t;
	in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
	cout << t;
	if (sizeof...(u)) cout << sep;
	out(u...);
}

struct IoSetupYamada {
	IoSetupYamada() {
		cin.tie(nullptr);
		ios::sync_with_stdio(false);
		cout << fixed << setprecision(15);
		cerr << fixed << setprecision(7);
	}
} iosetupyamada;

}  // namespace yamada
// End include: "inout.hpp"
// Begin include: "macro.hpp"
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define each3(x, y, z, v) for (auto&& [x, y, z] : v)
#define all(v) (v).begin(), (v).end()

#define rep1(a) for (long long _ = 0; _ < (long long)(a); ++_)
#define rep2(i, a) for (long long i = 0; i < (long long)(a); ++i)
#define rep3(i, a, b) for (long long i = a; i < (long long)(b); ++i)
#define rep4(i, a, b, c) for (long long i = a; i < (long long)(b); i += c)
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)

#define rep1r(a) for (long long i = (long long)(a)-1; i >= 0LL; --i)
#define rep2r(i, a) for (long long i = (long long)(a)-1; i >= 0LL; --i)
#define rep3r(i, a, b) for (long long i = (long long)(b)-1; i >= (long long)(a); --i)
#define overload3(a, b, c, d, ...) d
#define repr(...) overload3(__VA_ARGS__, rep3r, rep2r, rep1r)(__VA_ARGS__)

#define eb emplace_back
#define mkp make_pair
#define mkt make_tuple
#define fi first
#define se second

#define vv(type, name, h, ...)  \
	vector<vector<type> > name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) \
	vector<vector<vector<type>>> name( \
			h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)  \
	vector<vector<vector<vector<type>>>> name( \
			a, vector<vector<vector<type>>>( \
				b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

#define ini(...)   \
	int __VA_ARGS__; \
	in(__VA_ARGS__)
#define inl(...)         \
	long long __VA_ARGS__; \
	in(__VA_ARGS__)
#define ins(...)      \
	string __VA_ARGS__; \
	in(__VA_ARGS__)
#define in2(s, t)                           \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i]);                         \
	}
#define in3(s, t, u)                        \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i], u[i]);                   \
	}
#define in4(s, t, u, v)                     \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i], u[i], v[i]);             \
	}
#define die(...)             \
	do {                       \
		yamada::out(__VA_ARGS__);\
		return;                  \
	} while (0)
// End include: "macro.hpp"

namespace yamada {
void solve();
}
int main() { yamada::solve(); }
// End include: "../../template/template.hpp"
// Begin include: "../../monoid/basic.hpp"

namespace Monoid {

template <typename Derived, typename T>
struct Common {
	constexpr Derived operator+(const Derived &b) const { Derived r = static_cast<const Derived&>(*this); return r += b; }
	constexpr Derived operator-(const Derived &b) const { Derived r = static_cast<const Derived&>(*this); return r -= b; }
	constexpr Derived operator-() const { return Derived() - static_cast<const Derived&>(*this); }
	constexpr operator T() const { return static_cast<const Derived&>(*this).a; }
	constexpr bool operator==(const Derived &b) const { return static_cast<const Derived&>(*this).a == b.a; }
	constexpr bool operator!=(const Derived &b) const { return static_cast<const Derived&>(*this).a != b.a; }
};

template <typename T>
struct Plus : Common<Plus<T>, T> {
	static constexpr bool COMMUTATIVE = 1;
	using value_type = T;
	T a;
	constexpr Plus &operator+=(const Plus &b) {
		a += T(b);
		return *this;
	}
	constexpr Plus &operator-=(const Plus &b) {
		a -= T(b);
		return *this;
	}
	constexpr Plus() : a() {}
	constexpr Plus(T _a) : a(_a) {}
};

template <typename T, T MINF>
struct Max : Common<Max<T, MINF>, T> {
	static constexpr bool COMMUTATIVE = 1;
	using value_type = T;
	T a;
	constexpr Max &operator+=(const Max &b) {
		a = std::max(a, T(b));
		return *this;
	}
	constexpr Max() : a(MINF) {}
	constexpr Max(T _a) : a(_a) {}
};

template <typename T, T INF>
struct Min : Common<Min<T, INF>, T> {
	static constexpr bool COMMUTATIVE = 1;
	using value_type = T;
	T a;
	constexpr Min &operator+=(const Min &b) {
		a = std::min(a, T(b));
		return *this;
	}
	constexpr Min() : a(INF) {}
	constexpr Min(T _a) : a(_a) {}
};

template <typename T, T UNUSED_VALUE>
struct Update : Common<Update<T, UNUSED_VALUE>, T> {
	static constexpr bool COMMUTATIVE = 0;
	using value_type = T;
	T a;
	constexpr Update &operator+=(const Update &b) {
		if (T(b) != UNUSED_VALUE) a = T(b);
		return *this;
	}
	constexpr Update() : a(UNUSED_VALUE) {}
	constexpr Update(T _a) : a(_a) {}
};

template <typename MonoidBase>
struct CntMonoid { // コンテスト中にこれを直接扱うのは、頭が爆発するので非推奨
	static constexpr bool COMMUTATIVE = MonoidBase::COMMUTATIVE;
	using value_type = typename std::pair<int, MonoidBase>;
	using T = value_type;
	using T_Base = typename MonoidBase::value_type;
	T a;
	constexpr CntMonoid &operator+=(const CntMonoid &b) {
		a.first += T(b).first;
		a.second += T(b).second;
		return *this;
	}
	constexpr CntMonoid() : a(0, MonoidBase()) {}
	constexpr CntMonoid(T _a) : a(_a) {}
	constexpr CntMonoid(T_Base _a) : a(1, _a) {}
	constexpr CntMonoid operator+(const CntMonoid &b) const { return CntMonoid(*this) += b; }
	constexpr operator T() const { return a; }
	constexpr operator T_Base() const { return T_Base(a.second); }
	constexpr bool operator==(const CntMonoid &b) const { return CntMonoid(*this).a == b.a; }
	constexpr bool operator!=(const CntMonoid &b) const { return CntMonoid(*this).a != b.a; }
};

template <typename mint>
struct Affine : Common<Affine<mint>, std::pair<mint, mint>> {
	static constexpr bool COMMUTE = 0;
	using value_type = std::pair<mint, mint>;
	using T = value_type;
	T a;
	constexpr Affine &operator+=(const Affine &b) {
		a.first *= T(b).first;
		a.second *= T(b).first;
		a.second += T(b).second;
		return *this;
	}
	constexpr Affine() : a(1, 0) {}
	constexpr Affine(T _a) : a(_a) {}
	constexpr Affine(mint fi, mint se) : a({fi, se}) {}
};

} // namespace Monoid

namespace ActedMonoid {

using namespace Monoid;

template <typename Derived, typename Base>
struct CRTP : Base {
	using Base::Base;
	constexpr Derived& operator+=(const Derived& b) { Base::operator+=(b); return static_cast<Derived&>(*this); };
	constexpr Derived operator+(const Derived& b) const { Derived r = static_cast<const Derived&>(*this); return r += b; }
	constexpr bool operator==(const Derived &b) const { return static_cast<const Derived&>(*this).a == b.a; }
	constexpr bool operator!=(const Derived &b) const { return static_cast<const Derived&>(*this).a != b.a; }
};

template <typename T, T MINF>
struct Add_Max : CRTP<Add_Max<T, MINF>, Max<T, MINF>> {
	using Base = CRTP<Add_Max<T, MINF>, Max<T, MINF>>;
	using Base::Base;
	using ActingMonoid = Plus<T>;
	constexpr Add_Max<T, MINF> &operator^=(const ActingMonoid &b) {
		this->a += T(b);
		return *this;
	}
	constexpr Add_Max<T, MINF> operator^(const ActingMonoid &b) { return Add_Max<T, MINF>(*this) ^= b; }
};

template <typename T, T INF>
struct Add_Min : CRTP<Add_Min<T, INF>, Min<T, INF>> {
	using Base = CRTP<Add_Min<T, INF>, Min<T, INF>>;
	using Base::Base;
	using ActingMonoid = Plus<T>;
	constexpr Add_Min<T, INF> &operator^=(const ActingMonoid &b) {
		this->a += T(b);
		return *this;
	}
	constexpr Add_Min<T, INF> operator^(const ActingMonoid &b) { return Add_Min<T, INF>(*this) ^= b; }
};

template <typename T>
struct Add_Sum : CRTP<Add_Sum<T>, CntMonoid<Plus<T>>> {
	using Base = CRTP<Add_Sum<T>, CntMonoid<Plus<T>>>;
	using Base::Base;
	using ActingMonoid = Plus<T>;
	constexpr Add_Sum<T> &operator^=(const ActingMonoid &b) {
		(this->a).second += T(b) * (this->a).first;
		return *this;
	}
	constexpr Add_Sum<T> operator^(const ActingMonoid &b) { return Add_Sum(*this) ^= b; }
};

template <typename T, T MINF>
struct Update_Max : CRTP<Update_Max<T, MINF>, Max<T, MINF>> {
	using Base = CRTP<Update_Max<T, MINF>, Max<T, MINF>>;
	using Base::Base;
	using ActingMonoid = Update<T, MINF>;
	constexpr Update_Max<T, MINF> &operator^=(const ActingMonoid &b) {
		if (T(b) != MINF) this->a = T(b);
		return *this;
	}
	constexpr Update_Max<T, MINF> operator^(const ActingMonoid &b) { return Update_Max<T, MINF>(*this) ^= b; }
};

template <typename T, T INF>
struct Update_Min : CRTP<Update_Min<T, INF>, Min<T, INF>> {
	using Base = CRTP<Update_Min<T, INF>, Min<T, INF>>;
	using Base::Base;
	using ActingMonoid = Update<T, INF>;
	constexpr Update_Min<T, INF> &operator^=(const ActingMonoid &b) {
		if (T(b) != INF) this->a = T(b);
		return *this;
	}
	constexpr Update_Min<T, INF> operator^(const ActingMonoid &b) { return Update_Min<T, INF>(*this) ^= b; }
};

template <typename T, T UNUSED_VALUE>
struct Update_Sum : CRTP<Update_Sum<T, UNUSED_VALUE>, CntMonoid<Plus<T>>> {
	using Base = CRTP<Update_Sum<T, UNUSED_VALUE>, CntMonoid<Plus<T>>>;
	using Base::Base;
	using ActingMonoid = Update<T, UNUSED_VALUE>;
	constexpr Update_Sum<T, UNUSED_VALUE> &operator^=(const ActingMonoid &b) {
		if (T(b) != UNUSED_VALUE) (this->a).second = T(b) * (this->a).first;
		return *this;
	}
	constexpr Update_Sum<T, UNUSED_VALUE> operator^(const ActingMonoid &b) { return Update_Sum<T, UNUSED_VALUE>(*this) ^= b; }
};

template <typename mint>
struct Affine_Sum : CRTP<Affine_Sum<mint>, CntMonoid<Plus<mint>>> {
	using Base = CRTP<Affine_Sum<mint>, CntMonoid<Plus<mint>>>;
	using Base::Base;
	using ActingMonoid = Affine<mint>;
	constexpr Affine_Sum<mint> &operator^=(const ActingMonoid &b) {
		((this->a).second).a *= b.a.first;
		((this->a).second).a += b.a.second * (this->a).first;
		return *this;
	}
	constexpr Affine_Sum<mint> operator^(const ActingMonoid &b) { return Affine_Sum<mint>(*this) ^= b; }
};

} // namespace ActedMonoid
// End include: "../../monoid/basic.hpp"
// Begin include: "../../segment-tree/dynamic-segment-tree.hpp"

// Begin include: "../data-structure/node-pool.hpp"

// マルチテストケースに弱いので static で確保すること
template <typename Node>
struct NodePool {
	struct Slot {
		union alignas(Node) {
			Slot* next;
			unsigned char storage[sizeof(Node)];
		};
	};
	using Ptr = Node*;

	static constexpr int CHUNK_SIZE = 1 << 12;

	std::vector<std::unique_ptr<Slot[]>> chunks;
	Slot* cur = nullptr;
	int cur_used = 0;
	Slot* free_head = nullptr;

	NodePool() { alloc_chunk(); }

	template <typename... Args>
	Ptr create(Args&&... args) {
		Slot* s = new_slot();
		return ::new (s) Node(std::forward<Args>(args)...);
	}

	Ptr clone(const Ptr x) {
		assert(x);
		Slot* s = new_slot();
		return ::new (s) Node(*x);  // コピーコンストラクタ呼び出し
	}

	void destroy(Ptr x) {
		if (!x) return;
		x->~Node();
		auto s = reinterpret_cast<Slot*>(x);
		s->next = free_head;
		free_head = s;
	}

	void reset() {
		free_head = nullptr;
		if (!chunks.empty()) {
			cur = chunks[0].get();
			cur_used = 0;
		}
	}

	private:
	void alloc_chunk() {
		chunks.emplace_back(std::make_unique<Slot[]>(CHUNK_SIZE));
		cur = chunks.back().get();
		cur_used = 0;
	}

	Slot* new_slot() {
		if (free_head) {
			Slot* s = free_head;
			free_head = free_head->next;
			return s;
		}
		if (cur_used == CHUNK_SIZE) alloc_chunk();
		return &cur[cur_used++];
	}
};
// End include: "../data-structure/node-pool.hpp"

template <typename AM, bool PERSISTENT = false>
struct DynamicLazySegmentTree {
	using S = AM;
	using F = typename S::ActingMonoid;
	using ll = long long;
	struct Node {
		Node *l, *r;
		S sum;
		F laz;
	};

	NodePool<Node> pool;
	const ll L0, R0;
	using Ptr = Node *;
	using I = function<S(ll, ll)>;
	I init;
	Ptr cur_root;

	DynamicLazySegmentTree(ll L0, ll R0, I init = [](ll, ll) -> S { return S(); })
		: L0(L0), R0(R0), init(init), cur_root(new_root()) {}
	template <typename T>
	DynamicLazySegmentTree(const std::vector<T> &dat)
		: L0(0), R0(dat.size()), init([](ll, ll) -> S { return S(); }), cur_root(my_new(dat)) {}

	S prod(Ptr root, ll l, ll r) {
		if (l == r || !root) return S();
		assert(L0 <= l && l < r && r <= R0);
		S sum = S();
		_prod(root, L0, R0, l, r, sum, F());
		return sum;
	}
	S prod(ll l, ll r) { return prod(cur_root, l, r); }
	S prod(Ptr root) { return root ? root->sum : S(); }
	S prod() { return prod(cur_root); }
	S get_val(ll i) { return get_val(cur_root, i); }
	F get_laz(ll i) { return get(cur_root, i); }
	Ptr set_val(ll i, const S &sum) { return set_val(cur_root, i, sum); }
	Ptr multiply(ll i, const S &sum) { return multiply(cur_root, i, sum); }
	Ptr apply(ll l, ll r, const F &laz) { return apply(cur_root, l, r, laz); }

	S get_val(Ptr root, ll i) {
		ll L = L0, R = R0;
		while (L + 1 < R) {
			push(root, L, R);
			ll M = (L + R) / 2;
			if (i < M) {
				if (!root->l) return init(i, i + 1);
				root = root->l, R = M;
			}
			else {
				if (!root->r) return init(i, i + 1);
				root = root->r, L = M;
			}
		}
		return root->sum;
	}

	F get_laz(Ptr root, ll i) {
		if (!root) return F();
		F laz = F();
		_get_laz(root, L0, R0, i, laz);
		return laz;
	}

	Ptr set_val(Ptr root, ll i, const S &sum) {
		assert(root && L0 <= i && i < R0);
		return cur_root = _set_val(root, L0, R0, i, sum);
	}

	Ptr multiply(Ptr root, ll i, const S &sum) {
		assert(root && L0 <= i && i < R0);
		return cur_root = _multiply(root, L0, R0, i, sum);
	}

	Ptr apply(Ptr root, ll l, ll r, const F &laz) {
		if (l == r) return root;
		assert(root && L0 <= l && l < r && r <= R0);
		return cur_root = _apply(root, L0, R0, l, r, laz);
	}

	template <typename G>
	ll max_right(Ptr root, G check, ll L) {
		assert(root && L0 <= L && L <= R0 && check(S()));
		S sum = S();
		return _max_right(root, check, L0, R0, L, sum);
	}

	template <typename G>
	ll min_left(Ptr root, G check, ll R) {
		assert(root && L0 <= R && R <= R0 && check(S()));
		S sum = S();
		return _min_left(root, check, L0, R0, R, sum);
	}

	// memo(idx, val)
	template <typename MEMO>
	void enumerate(Ptr root, MEMO memo) {
		auto dfs = [&](auto &dfs, Ptr u, ll l, ll r, F laz) -> void {
			if (!u) return;
			if (r - l == 1) {
				memo(l, u->sum ^ laz);
				return;
			}
			ll m = (l + r) / 2;
			laz = u->laz + laz;
			dfs(dfs, u->l, l, m, laz);
			dfs(dfs, u->r, m, r, laz);
		};
		dfs(dfs, root, L0, R0, F());
	}

	// u[l:r) を apply(v[l:r), laz) で上書きしたものを返す
	Ptr copy_interval(Ptr u, Ptr v, ll l, ll r, F laz = F()) {
		static_assert(PERSISTENT);
		if (u == v) return u;
		u = clone(u);
		_copy_interval(u, v, L0, R0, l, r, laz);
		return u;
	}

	private:
	Ptr new_root() { return my_new(L0, R0); }

	Ptr my_new(const S sum) {
		Ptr u = pool.create();
		u->l = u->r = nullptr;
		u->sum = sum, u->laz = F();
		return u;
	}

	Ptr my_new(ll l, ll r) { return my_new(init(l, r)); }
	Ptr my_new() { return my_new(L0, R0); }
	void my_del(Ptr u) { delete u; }

	Ptr clone(Ptr u) {
		if constexpr (!PERSISTENT) return u;
		if (!u) return u;
		return pool.clone(u);
	}

	void push(Ptr u, ll l, ll r) {
		assert(r - l >= 2);
		ll m = (l + r) / 2;
		if (u->laz == F()) return;
		u->l = (u->l ? clone(u->l) : my_new(l, m));
		u->l->sum ^= u->laz;
		u->l->laz += u->laz;
		u->r = (u->r ? clone(u->r) : my_new(m, r));
		u->r->sum ^= u->laz;
		u->r->laz += u->laz;
		u->laz = F();
	}

	void _copy_interval(Ptr u, Ptr v, ll l, ll r, ll ql, ll qr, F laz) {
		// u[ql, qr) <- apply(v[ql, qr), laz)
		// u is already updated
		assert(u);
		ql = std::max(ql, l);
		qr = std::min(qr, r);
		if (ql >= qr) return;
		if (l == ql && r == qr) { // terminus
			if (v) {
				u->sum = v->sum ^ laz, u->laz = v->laz + laz;
				u->l = v->l, u->r = v->r;
			}
			else {
				u->sum = init(l, r) ^ laz, u->laz = laz;
				u->l = nullptr, u->r = nullptr;
			}
			return;
		}
		// push
		ll m = (l + r) / 2;
		u->l = (u->l ? clone(u->l) : my_new());
		u->r = (u->r ? clone(u->r) : my_new());
		u->l->sum ^= u->laz;
		u->l->laz += u->laz;
		u->r->sum ^= u->laz;
		u->r->laz += u->laz;
		u->laz = F();
		if (v) laz = v->laz + laz;
		_copy_interval(u->l, (v && v->l ? v->l : nullptr), l, m, ql, qr, laz);
		_copy_interval(u->r, (v && v->r ? v->r : nullptr), m, r, ql, qr, laz);
		u->sum = u->l->sum + u->r->sum;
	}

	template <typename T>
	Ptr my_new(const std::vector<T> &dat) {
		assert(L0 == 0 && R0 == dat.size());
		auto dfs = [&](auto &dfs, ll l, ll r) -> Ptr {
			if (l == r) return nullptr;
			if (r == l + 1) return my_new(S(dat[l]));
			ll m = (l + r) / 2;
			Ptr l_root = dfs(dfs, l, m), r_root = dfs(dfs, m, r);
			S sum = l_root->sum + r_root->sum;
			Ptr cur_root = my_new(sum);
			cur_root->l = l_root, cur_root->r = r_root;
			return cur_root;
		};
		return dfs(dfs, 0, dat.size());
	}

	Ptr _set_val(Ptr u, ll l, ll r, ll i, const S &sum) {
		if (r == l + 1) {
			u = clone(u);
			u->sum = sum;
			u->laz = F();
			return u;
		}
		push(u, l, r);
		ll m = (l + r) / 2;
		if (!u->l) u->l = my_new(l, m);
		if (!u->r) u->r = my_new(m, r);

		u = clone(u);
		if (i < m) {
			u->l = _set_val(u->l, l, m, i, sum);
		} else {
			u->r = _set_val(u->r, m, r, i, sum);
		}
		u->sum = u->l->sum + u->r->sum;
		return u;
	}

	Ptr _multiply(Ptr u, ll l, ll r, ll i, const S &sum) {
		if (r == l + 1) {
			u = clone(u);
			u->sum += sum;
			u->laz = F();
			return u;
		}
		push(u, l, r);
		ll m = (l + r) / 2;
		if (!u->l) u->l = my_new(l, m);
		if (!u->r) u->r = my_new(m, r);

		u = clone(u);
		if (i < m) {
			u->l = _multiply(u->l, l, m, i, sum);
		} else {
			u->r = _multiply(u->r, m, r, i, sum);
		}
		u->sum = u->l->sum + u->r->sum;
		return u;
	}

	void _prod(Ptr u, ll l, ll r, ll ql, ll qr, S &sum, F laz) {
		ql = std::max(ql, l);
		qr = std::min(qr, r);
		if (ql >= qr) return;
		if (!u) {
			sum += (init(ql, qr) ^ laz);
			return;
		}
		if (l == ql && r == qr) {
			sum += (u->sum ^ laz);
			return;
		}
		ll m = (l + r) / 2;
		laz = u->laz + laz;
		_prod(u->l, l, m, ql, qr, sum, laz);
		_prod(u->r, m, r, ql, qr, sum, laz);
	}

	Ptr _apply(Ptr u, ll l, ll r, ll ql, ll qr, const F &laz) {
		if (!u) u = my_new(l, r);
		ql = std::max(ql, l);
		qr = std::min(qr, r);
		if (ql >= qr) return u;
		if (l == ql && r == qr) {
			u = clone(u);
			u->sum ^= laz;
			u->laz += laz;
			return u;
		}
		push(u, l, r);
		ll m = (l + r) / 2;
		u = clone(u);
		u->l = _apply(u->l, l, m, ql, qr, laz);
		u->r = _apply(u->r, m, r, ql, qr, laz);
		u->sum = u->l->sum + u->r->sum;
		return u;
	}

	template <typename G>
	ll _max_right(Ptr u, const G &check, ll l, ll r, ll ql, S &sum) {
		if (r <= ql) return r;
		if (!u) u = my_new(l, r);
		ql = std::max(ql, l);
		if (l == ql && check(sum + u->sum)) {
			sum += u->sum;
			return r;
		}
		if (r == l + 1) return l;
		push(u, l, r);
		ll m = (l + r) / 2;
		ll k = _max_right(u->l, check, l, m, ql, sum);
		if (k < m) return k;
		return _max_right(u->r, check, m, r, ql, sum);
	}

	template <typename G>
	ll _min_left(Ptr u, const G &check, ll l, ll r, ll qr, S &sum) {
		if (qr <= l) return l;
		if (!u) u = my_new(l, r);
		qr = std::min(qr, r);
		if (r == qr && check(u->sum + sum)) {
			sum = u->sum + sum;
			return l;
		}
		if (r == l + 1) return r;
		push(u, l, r);
		ll m = (l + r) / 2;
		ll k = _min_left(u->r, check, m, r, qr, sum);
		if (m < k) return k;
		return _min_left(u->l, check, l, m, qr, sum);
	}

	void _get_laz(Ptr u, ll l, ll r, ll i, F &laz) {
		if (!u) return;
		laz = u->laz + laz;
		if (r == l + 1) return;
		ll m = (l + r) / 2;
		if (i < m) return _get_laz(u->l, l, m, i, laz);
		_get_laz(u->r, m, r, i, laz);
	}
};
// End include: "../../segment-tree/dynamic-segment-tree.hpp"
// Begin include: "../../segment-tree/lazy-segment-tree-utility.hpp"

// Begin include: "../monoid/basic.hpp"

// End include: "../monoid/basic.hpp"
// Begin include: "lazy-segment-tree.hpp"

template <typename AM>
struct LazySegmentTree {
	using S = AM;
	using F = typename AM::ActingMonoid;
	int N, LOG, M;
	std::vector<S> dat; // このライブラリはlaz反映済式
	std::vector<F> laz;

	LazySegmentTree() {}
	LazySegmentTree(int N) { build(N); }
	template <typename I>
	LazySegmentTree(int N, I init) { build(N, init); }
	template <typename U>
	LazySegmentTree(const std::vector<U>& v) { build(v); }

	void set(int p, S x) {
		assert(0 <= p && p < N);
		p += M;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		dat[p] = x;
		for (int i = 1; i <= LOG; i++) update(p >> i);
	}
	void multiply(int p, const S& x) {
		assert(0 <= p && p < N);
		p += M;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		dat[p] += x;
		for (int i = 1; i <= LOG; i++) update(p >> i);
	}

	S get(int p) {
		assert(0 <= p && p < N);
		p += M;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		return dat[p];
	}
	std::vector<S> get() {
		for (int k = 1; k < M; k++) push(k);
		return {dat.begin() + M, dat.begin() + M + N};
	}
	S operator[](const int &k) { return get(k); }

	S prod(int l, int r) {
		assert(0 <= l && l <= r && r <= N);
		if (l == r) return S();
		l += M, r += M;
		for (int i = LOG; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push((r - 1) >> i);
		}
		S xl, xr;
		while (l < r) {
			if (l & 1) xl = xl + dat[l++];
			if (r & 1) xr = dat[--r] + xr;
			l >>= 1, r >>= 1;
		}
		return xl + xr;
	}
	S prod() { return dat[1]; }
	inline S operator()(int l, int r) { return prod(l, r); }

	void apply(int l, int r, F f) {
		assert(0 <= l && l <= r && r <= N);
		if (l == r) return;
		l += M, r += M;
		for (int i = LOG; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push((r - 1) >> i);
		}
		int l2 = l, r2 = r;
		while (l < r) {
			if (l & 1) apply_at(l++, f);
			if (r & 1) apply_at(--r, f);
			l >>= 1, r >>= 1;
		}
		l = l2, r = r2;
		for (int i = 1; i <= LOG; i++) {
			if (((l >> i) << i) != l) update(l >> i);
			if (((r >> i) << i) != r) update((r - 1) >> i);
		}
	}

	template <typename G>
	int max_right(int l, const G check) {
		assert(0 <= l && l <= N);
		assert(check(S()));
		if (l == N) return N;
		l += M;
		for (int i = LOG; i >= 1; i--) push(l >> i);
		S sm;
		do {
			while (l % 2 == 0) l >>= 1;
			if (!check(sm + dat[l])) {
				while (l < M) {
					push(l);
					l = (2 * l);
					if (check(sm + dat[l])) sm += dat[l++];
				}
				return l - M;
			}
			sm += dat[l++];
		} while ((l & -l) != l);
		return N;
	}

	template <typename G>
	int min_left(int r, const G check) {
		assert(0 <= r && r <= N);
		assert(check(S()));
		if (r == 0) return 0;
		r += M;
		for (int i = LOG; i >= 1; i--) push((r - 1) >> i);
		S sm;
		do {
			r--;
			while (r > 1 && (r % 2)) r >>= 1;
			if (!check(dat[r] + sm)) {
				while (r < M) {
					push(r);
					r = (2 * r + 1);
					if (check(dat[r] + sm)) sm = dat[r--] + sm;
				}
				return r + 1 - M;
			}
			sm = dat[r] + sm;
		} while ((r & -r) != r);
		return 0;
	}

	private:
	void update(int k) { dat[k] = dat[k * 2] + dat[k * 2 + 1]; }
	void apply_at(int k, F f) {
		dat[k] ^= f;
		if (k < M) laz[k] += f;
	}
	void push(int k) {
		if (laz[k] == F()) return;
		apply_at(2 * k, laz[k]), apply_at(2 * k + 1, laz[k]);
		laz[k] = F();
	}

	void build(int _N) { build(_N, [](int i) -> S { return S(); }); }
	template <typename U>
	void build(const std::vector<U>& v) { build(v.size(), [&](int i) -> S { return S(v[i]); }); }
	template <typename I>
	void build(int _N, I init) {
		N = _N, LOG = 1;
		while ((1 << LOG) < N) ++LOG;
		M = 1 << LOG;
		dat.assign(M << 1, S());
		laz.assign(M, F());
		for (int i = 0; i < N; i++) dat[M + i] = init(i);
		for (int i = M - 1; i >= 1; i--) update(i);
	}
};
// End include: "lazy-segment-tree.hpp"

template <typename T, T MINF>
using AddMax_LazySegmentTree = LazySegmentTree<ActedMonoid::Add_Max<T, MINF>>;
template <typename T, T INF>
using AddMin_LazySegmentTree = LazySegmentTree<ActedMonoid::Add_Min<T, INF>>;
template <typename T>
using AddSum_LazySegmentTree = LazySegmentTree<ActedMonoid::Add_Sum<T>>;
template <typename T, T MINF>
using UpdateMax_LazySegmentTree = LazySegmentTree<ActedMonoid::Update_Max<T, MINF>>;
template <typename T, T INF>
using UpdateMin_LazySegmentTree = LazySegmentTree<ActedMonoid::Update_Min<T, INF>>;
template <typename T, T UNUSED_VALUE>
using UpdateSum_LazySegmentTree = LazySegmentTree<ActedMonoid::Update_Sum<T, UNUSED_VALUE>>;
template <typename mint>
using AffineSum_LazySegmentTree = LazySegmentTree<ActedMonoid::Affine_Sum<mint>>;

// End include: "../../segment-tree/lazy-segment-tree-utility.hpp"
using namespace yamada;

void tc()
{
	inl(N,M);
	vp AB(M); in(AB);
	each2(a,b,AB)--a;
	if(N==M)die("Yes");
	{
		set<ll> st;
		each2(a,b,AB)st.insert(a);
		ll ofst=-1;
		each2(a,b,AB){
			ll a2=(a+N-1)%N;
			if(st.find(a2)==st.end()){
				ofst=a;
				break;
			}
		}
		assert(ofst>=0);
		each2(a,b,AB)a=(a+N-ofst)%N;
		sort(all(AB));
	}
	vl A,B;
	each2(a,b,AB)A.eb(a),B.eb(b);
	auto Brui=mkrui(B);

	vp lr;
	for(ll l=0,r=0; l<M; l=r){
		while(r<M && AB[r].fi-r==AB[l].fi-l)++r;
		lr.eb(A[l],A[r-1]+1);
	}
	{
		vp LR;
		rep(i,lr.size()){
			LR.eb(lr[i]);
			if(i<lr.size()-1){
				LR.eb(lr[i].se,lr[i+1].fi);
			}
		}
		LR.eb(lr.back().se,N);
		lr=LR;
	}
	ll K=lr.size();

	// 左にはみ出す
	DynamicLazySegmentTree<ActedMonoid::Update_Max<ll,-infLL>> seg(0,N,[](ll l,ll r)->ll{return 1;});
	rep(k,0,K,2){
		auto[l,r]=lr[k];
		auto[l1,r1]=lr[(k+K-1)%K];
		auto[l2,r2]=lr[(k+1)%K];
		ll len1=r1-l1;
		ll len2=r2-l2;
		ll aiL=lb(A,l);
		ll aiR=lb(A,r-1);

		ll L=-infLL,R=infLL;
		if(r-l>=2){
			rep(i,l+1,r-1){
				ll ai=lb(A,i);
				if(B[ai]!=1)die("No");
			}
			if(B[aiL]-1>len1)die("No");
			if(B[aiR]-1>len2)die("No");

			amax(L,B[aiL]-1);
			amin(R,B[aiL]-1);
		}
		else{
			ll aiL=lb(A,l);
			ll b=B[aiL];

			amax(L,0);
			amin(R,b-1);
			amin(R,len1);
			amax(L,b-len2-1);
		}
		L+=l-Brui[aiL];
		R+=l-Brui[aiL];
		amax(R,-1); amin(R,N-1);
		amax(L,0); amin(L,N);
		seg.apply(0,L,0);
		seg.apply(R+1,N,0);
	}

	YesNo(seg.prod().a);
}
void yamada::solve() { ini(T); while(T--)tc(); }