// 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: "../../internal/internal-math.hpp"

// Begin include: "internal-type-traits.hpp"

#include <type_traits>
using namespace std;

namespace internal {
template <typename T>
using is_broadly_integral =
    typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> ||
                               is_same_v<T, __uint128_t>,
                           true_type, false_type>::type;

template <typename T>
using is_broadly_signed =
    typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>,
                           true_type, false_type>::type;

template <typename T>
using is_broadly_unsigned =
    typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>,
                           true_type, false_type>::type;

#define ENABLE_VALUE(x) \
  template <typename T> \
  constexpr bool x##_v = x<T>::value;

ENABLE_VALUE(is_broadly_integral);
ENABLE_VALUE(is_broadly_signed);
ENABLE_VALUE(is_broadly_unsigned);
#undef ENABLE_VALUE

#define ENABLE_HAS_TYPE(var)                                   \
  template <class, class = void>                               \
  struct has_##var : false_type {};                            \
  template <class T>                                           \
  struct has_##var<T, void_t<typename T::var>> : true_type {}; \
  template <class T>                                           \
  constexpr auto has_##var##_v = has_##var<T>::value;

#define ENABLE_HAS_VAR(var)                                     \
  template <class, class = void>                                \
  struct has_##var : false_type {};                             \
  template <class T>                                            \
  struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \
  template <class T>                                            \
  constexpr auto has_##var##_v = has_##var<T>::value;

}  // namespace internal
// End include: "internal-type-traits.hpp"

namespace internal {

#include <cassert>
#include <utility>
#include <vector>
using namespace std;

// a mod p
template <typename T>
T safe_mod(T a, T p) {
  a %= p;
  if constexpr (is_broadly_signed_v<T>) {
    if (a < 0) a += p;
  }
  return a;
}

// 返り値：pair(g, x)
// s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
template <typename T>
pair<T, T> inv_gcd(T a, T p) {
  static_assert(is_broadly_signed_v<T>);
  a = safe_mod(a, p);
  if (a == 0) return {p, 0};
  T b = p, x = 1, y = 0;
  while (a != 0) {
    T q = b / a;
    swap(a, b %= a);
    swap(x, y -= q * x);
  }
  if (y < 0) y += p / b;
  return {b, y};
}

// 返り値 : a^{-1} mod p
// gcd(a, p) != 1 が必要
template <typename T>
T inv(T a, T p) {
  static_assert(is_broadly_signed_v<T>);
  a = safe_mod(a, p);
  T b = p, x = 1, y = 0;
  while (a != 0) {
    T q = b / a;
    swap(a, b %= a);
    swap(x, y -= q * x);
  }
  assert(b == 1);
  return y < 0 ? y + p : y;
}

// T : 底の型
// U : T*T がオーバーフローしない かつ 指数の型
template <typename T, typename U>
T modpow(T a, U n, T p) {
  a = safe_mod(a, p);
  T ret = 1 % p;
  while (n != 0) {
    if (n % 2 == 1) ret = U(ret) * a % p;
    a = U(a) * a % p;
    n /= 2;
  }
  return ret;
}

// 返り値 : pair(rem, mod)
// 解なしのときは {0, 0} を返す
template <typename T>
pair<T, T> crt(const vector<T>& r, const vector<T>& m) {
  static_assert(is_broadly_signed_v<T>);
  assert(r.size() == m.size());
  int n = int(r.size());
  T r0 = 0, m0 = 1;
  for (int i = 0; i < n; i++) {
    assert(1 <= m[i]);
    T r1 = safe_mod(r[i], m[i]), m1 = m[i];
    if (m0 < m1) swap(r0, r1), swap(m0, m1);
    if (m0 % m1 == 0) {
      if (r0 % m1 != r1) return {0, 0};
      continue;
    }
    auto [g, im] = inv_gcd(m0, m1);
    T u1 = m1 / g;
    if ((r1 - r0) % g) return {0, 0};
    T x = (r1 - r0) / g % u1 * im % u1;
    r0 += x * m0;
    m0 *= u1;
    if (r0 < 0) r0 += m0;
  }
  return {r0, m0};
}

}  // namespace internal
// End include: "../../internal/internal-math.hpp"
// Begin include: "../../atcoder/math.hpp"
#ifndef ATCODER_MATH_HPP
#define ATCODER_MATH_HPP 1

#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>
// Begin include: "internal_math.hpp"
#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1

#include <utility>

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m < 2^31`
    barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

}  // namespace internal

}  // namespace atcoder

#endif  // ATCODER_INTERNAL_MATH_HPP
// End include: "internal_math.hpp"

namespace atcoder {

long long pow_mod(long long x, long long n, int m) {
    assert(0 <= n && 1 <= m);
    if (m == 1) return 0;
    internal::barrett bt((unsigned int)(m));
    unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
    while (n) {
        if (n & 1) r = bt.mul(r, y);
        y = bt.mul(y, y);
        n >>= 1;
    }
    return r;
}

long long inv_mod(long long x, long long m) {
    assert(1 <= m);
    auto z = internal::inv_gcd(x, m);
    assert(z.first == 1);
    return z.second;
}

// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long>& r,
                                    const std::vector<long long>& m) {
    assert(r.size() == m.size());
    int n = int(r.size());
    // Contracts: 0 <= r0 < m0
    long long r0 = 0, m0 = 1;
    for (int i = 0; i < n; i++) {
        assert(1 <= m[i]);
        long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
        if (m0 < m1) {
            std::swap(r0, r1);
            std::swap(m0, m1);
        }
        if (m0 % m1 == 0) {
            if (r0 % m1 != r1) return {0, 0};
            continue;
        }
        // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

        // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
        // r2 % m0 = r0
        // r2 % m1 = r1
        // -> (r0 + x*m0) % m1 = r1
        // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
        // -> x = (r1 - r0) / g * inv(u0) (mod u1)

        // im = inv(u0) (mod u1) (0 <= im < u1)
        long long g, im;
        std::tie(g, im) = internal::inv_gcd(m0, m1);

        long long u1 = (m1 / g);
        // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
        if ((r1 - r0) % g) return {0, 0};

        // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
        long long x = (r1 - r0) / g % u1 * im % u1;

        // |r0| + |m0 * x|
        // < m0 + m0 * (u1 - 1)
        // = m0 + m0 * m1 / g - m0
        // = lcm(m0, m1)
        r0 += x * m0;
        m0 *= u1;  // -> lcm(m0, m1)
        if (r0 < 0) r0 += m0;
    }
    return {r0, m0};
}

long long floor_sum(long long n, long long m, long long a, long long b) {
    long long ans = 0;
    if (a >= m) {
        ans += (n - 1) * n * (a / m) / 2;
        a %= m;
    }
    if (b >= m) {
        ans += n * (b / m);
        b %= m;
    }

    long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
    if (y_max == 0) return ans;
    ans += (n - (x_max + a - 1) / a) * y_max;
    ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
    return ans;
}

}  // namespace atcoder

#endif  // ATCODER_MATH_HPP
// End include: "../../atcoder/math.hpp"
// Begin include: "../../prime/fast-factorize.hpp"

#include <cstdint>
#include <numeric>
#include <vector>
using namespace std;

// Begin include: "../internal/internal-math.hpp"

// End include: "../internal/internal-math.hpp"
// Begin include: "../misc/rng.hpp"

// Begin include: "../internal/internal-seed.hpp"

#include <chrono>
using namespace std;

namespace internal {
unsigned long long non_deterministic_seed() {
  unsigned long long m =
      chrono::duration_cast<chrono::nanoseconds>(
          chrono::high_resolution_clock::now().time_since_epoch())
          .count();
  m ^= 9845834732710364265uLL;
  m ^= m << 24, m ^= m >> 31, m ^= m << 35;
  return m;
}
unsigned long long deterministic_seed() { return 88172645463325252UL; }

// 64 bit の seed 値を生成 (手元では seed 固定)
// 連続で呼び出すと同じ値が何度も返ってくるので注意
// #define RANDOMIZED_SEED するとシードがランダムになる
unsigned long long seed() {
#if defined(NyaanLocal) && !defined(RANDOMIZED_SEED)
  return deterministic_seed();
#else
  return non_deterministic_seed();
#endif
}

}  // namespace internal
// End include: "../internal/internal-seed.hpp"

namespace my_rand {
using i64 = long long;
using u64 = unsigned long long;

// [0, 2^64 - 1)
u64 rng() {
	static u64 _x = internal::seed();
	return _x ^= _x << 7, _x ^= _x >> 9;
}

// [l, r]
i64 rng(i64 l, i64 r) {
	assert(l <= r);
	return l + rng() % u64(r - l + 1);
}

// [l, r)
i64 randint(i64 l, i64 r) {
	assert(l < r);
	return l + rng() % u64(r - l);
}

// choose N numbers from [l, r) without overlapping
vector<i64> randset(i64 l, i64 r, i64 N) {
	assert(l <= r && N <= r - l);
	unordered_set<i64> s;
	for (i64 i = N; i; --i) {
		i64 m = randint(l, r + 1 - i);
		if (s.find(m) != s.end()) m = r - i;
		s.insert(m);
	}
	vector<i64> ret;
	for (auto& x : s) ret.push_back(x);
	sort(begin(ret), end(ret));
	return ret;
}

// [0.0, 1.0)
double rnd() { return rng() * 5.42101086242752217004e-20; }
// [l, r)
double rnd(double l, double r) {
	assert(l < r);
	return l + rnd() * (r - l);
}

template <typename T>
void randshf(vector<T>& v) {
	int N = v.size();
	for (int i = 1; i < N; i++) swap(v[i], v[randint(0, i + 1)]);
}

}  // namespace my_rand

using my_rand::randint;
using my_rand::randset;
using my_rand::randshf;
using my_rand::rnd;
using my_rand::rng;
// End include: "../misc/rng.hpp"
// Begin include: "../modint/arbitrary-montgomery-modint.hpp"

#include <iostream>
using namespace std;

template <typename Int, typename UInt, typename Long, typename ULong, int id>
struct ArbitraryLazyMontgomeryModIntBase {
  using mint = ArbitraryLazyMontgomeryModIntBase;

  inline static UInt mod;
  inline static UInt r;
  inline static UInt n2;
  static constexpr int bit_length = sizeof(UInt) * 8;

  static UInt get_r() {
    UInt ret = mod;
    while (mod * ret != 1) ret *= UInt(2) - mod * ret;
    return ret;
  }
  static void set_mod(UInt m) {
    assert(m < (UInt(1u) << (bit_length - 2)));
    assert((m & 1) == 1);
    mod = m, n2 = -ULong(m) % m, r = get_r();
  }
  UInt a;

  ArbitraryLazyMontgomeryModIntBase() : a(0) {}
  ArbitraryLazyMontgomeryModIntBase(const Long &b)
      : a(reduce(ULong(b % mod + mod) * n2)){};

  static UInt reduce(const ULong &b) {
    return (b + ULong(UInt(b) * UInt(-r)) * mod) >> bit_length;
  }

  mint &operator+=(const mint &b) {
    if (Int(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator-=(const mint &b) {
    if (Int(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator*=(const mint &b) {
    a = reduce(ULong(a) * b.a);
    return *this;
  }
  mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  mint operator+(const mint &b) const { return mint(*this) += b; }
  mint operator-(const mint &b) const { return mint(*this) -= b; }
  mint operator*(const mint &b) const { return mint(*this) *= b; }
  mint operator/(const mint &b) const { return mint(*this) /= b; }

  bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  mint operator-() const { return mint(0) - mint(*this); }
  mint operator+() const { return mint(*this); }

  mint pow(ULong n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul, n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    Long t;
    is >> t;
    b = ArbitraryLazyMontgomeryModIntBase(t);
    return (is);
  }

  mint inverse() const {
    Int x = get(), y = get_mod(), u = 1, v = 0;
    while (y > 0) {
      Int t = x / y;
      swap(x -= t * y, y);
      swap(u -= t * v, v);
    }
    return mint{u};
  }

  UInt get() const {
    UInt ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static UInt get_mod() { return mod; }
};

// id に適当な乱数を割り当てて使う
template <int id>
using ArbitraryLazyMontgomeryModInt =
    ArbitraryLazyMontgomeryModIntBase<int, unsigned int, long long,
                                      unsigned long long, id>;
template <int id>
using ArbitraryLazyMontgomeryModInt64bit =
    ArbitraryLazyMontgomeryModIntBase<long long, unsigned long long, __int128_t,
                                      __uint128_t, id>;
// End include: "../modint/arbitrary-montgomery-modint.hpp"
// Begin include: "miller-rabin.hpp"

#include <vector>
using namespace std;

// Begin include: "../internal/internal-math.hpp"

// End include: "../internal/internal-math.hpp"
// Begin include: "../modint/arbitrary-montgomery-modint.hpp"

// End include: "../modint/arbitrary-montgomery-modint.hpp"

namespace fast_factorize {

template <typename T, typename U>
bool miller_rabin(const T& n, vector<T> ws) {
  if (n <= 2) return n == 2;
  if (n % 2 == 0) return false;

  T d = n - 1;
  while (d % 2 == 0) d /= 2;
  U e = 1, rev = n - 1;
  for (T w : ws) {
    if (w % n == 0) continue;
    T t = d;
    U y = internal::modpow<T, U>(w, t, n);
    while (t != n - 1 && y != e && y != rev) y = y * y % n, t *= 2;
    if (y != rev && t % 2 == 0) return false;
  }
  return true;
}

bool miller_rabin_u64(unsigned long long n) {
  return miller_rabin<unsigned long long, __uint128_t>(
      n, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
}

template <typename mint>
bool miller_rabin(unsigned long long n, vector<unsigned long long> ws) {
  if (n <= 2) return n == 2;
  if (n % 2 == 0) return false;

  if (mint::get_mod() != n) mint::set_mod(n);
  unsigned long long d = n - 1;
  while (~d & 1) d >>= 1;
  mint e = 1, rev = n - 1;
  for (unsigned long long w : ws) {
    if (w % n == 0) continue;
    unsigned long long t = d;
    mint y = mint(w).pow(t);
    while (t != n - 1 && y != e && y != rev) y *= y, t *= 2;
    if (y != rev && t % 2 == 0) return false;
  }
  return true;
}

bool is_prime(unsigned long long n) {
  using mint32 = ArbitraryLazyMontgomeryModInt<96229631>;
  using mint64 = ArbitraryLazyMontgomeryModInt64bit<622196072>;

  if (n <= 2) return n == 2;
  if (n % 2 == 0) return false;
  if (n < (1uLL << 30)) {
    return miller_rabin<mint32>(n, {2, 7, 61});
  } else if (n < (1uLL << 62)) {
    return miller_rabin<mint64>(
        n, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
  } else {
    return miller_rabin_u64(n);
  }
}

}  // namespace fast_factorize

using fast_factorize::is_prime;

/**
 * @brief Miller-Rabin primality test
 */
// End include: "miller-rabin.hpp"

namespace fast_factorize {
using u64 = uint64_t;

template <typename mint, typename T>
T pollard_rho(T n) {
  if (~n & 1) return 2;
  if (is_prime(n)) return n;
  if (mint::get_mod() != n) mint::set_mod(n);
  mint R, one = 1;
  auto f = [&](mint x) { return x * x + R; };
  auto rnd_ = [&]() { return rng() % (n - 2) + 2; };
  while (1) {
    mint x, y, ys, q = one;
    R = rnd_(), y = rnd_();
    T g = 1;
    constexpr int m = 128;
    for (int r = 1; g == 1; r <<= 1) {
      x = y;
      for (int i = 0; i < r; ++i) y = f(y);
      for (int k = 0; g == 1 && k < r; k += m) {
        ys = y;
        for (int i = 0; i < m && i < r - k; ++i) q *= x - (y = f(y));
        g = gcd(q.get(), n);
      }
    }
    if (g == n) do
        g = gcd((x - (ys = f(ys))).get(), n);
      while (g == 1);
    if (g != n) return g;
  }
  exit(1);
}

using i64 = long long;

vector<i64> inner_factorize(u64 n) {
  using mint32 = ArbitraryLazyMontgomeryModInt<452288976>;
  using mint64 = ArbitraryLazyMontgomeryModInt64bit<401243123>;

  if (n <= 1) return {};
  u64 p;
  if (n <= (1LL << 30)) {
    p = pollard_rho<mint32, uint32_t>(n);
  } else if (n <= (1LL << 62)) {
    p = pollard_rho<mint64, uint64_t>(n);
  } else {
    exit(1);
  }
  if (p == n) return {i64(p)};
  auto l = inner_factorize(p);
  auto r = inner_factorize(n / p);
  copy(begin(r), end(r), back_inserter(l));
  return l;
}

vector<i64> factorize(u64 n) {
  auto ret = inner_factorize(n);
  sort(begin(ret), end(ret));
  return ret;
}

map<i64, i64> factor_count(u64 n) {
  map<i64, i64> mp;
  for (auto &x : factorize(n)) mp[x]++;
  return mp;
}

vector<i64> divisors(u64 n) {
  if (n == 0) return {};
  vector<pair<i64, i64>> v;
  for (auto &p : factorize(n)) {
    if (v.empty() || v.back().first != p) {
      v.emplace_back(p, 1);
    } else {
      v.back().second++;
    }
  }
  vector<i64> ret;
  auto f = [&](auto rc, int i, i64 x) -> void {
    if (i == (int)v.size()) {
      ret.push_back(x);
      return;
    }
    rc(rc, i + 1, x);
    for (int j = 0; j < v[i].second; j++) rc(rc, i + 1, x *= v[i].first);
  };
  f(f, 0, 1);
  sort(begin(ret), end(ret));
  return ret;
}

}  // namespace fast_factorize

using fast_factorize::divisors;
using fast_factorize::factor_count;
using fast_factorize::factorize;

/**
 * @brief 高速素因数分解(Miller Rabin/Pollard's Rho)
 * @docs docs/prime/fast-factorize.md
 */
// End include: "../../prime/fast-factorize.hpp"
// Begin include: "../../modint/arbitrary-modint.hpp"

// Begin include: "barrett-reduction.hpp"

#include <utility>
using namespace std;

struct Barrett {
  using u32 = unsigned int;
  using i64 = long long;
  using u64 = unsigned long long;
  u32 m;
  u64 im;
  Barrett() : m(), im() {}
  Barrett(int n) : m(n), im(u64(-1) / m + 1) {}
  constexpr inline i64 quo(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    return m <= r ? x - 1 : x;
  }
  constexpr inline i64 rem(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    return m <= r ? r + m : r;
  }
  constexpr inline pair<i64, int> quorem(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    if (m <= r) return {x - 1, r + m};
    return {x, r};
  }
  constexpr inline i64 pow(u64 n, i64 p) {
    u32 a = rem(n), r = m == 1 ? 0 : 1;
    while (p) {
      if (p & 1) r = rem(u64(r) * a);
      a = rem(u64(a) * a);
      p >>= 1;
    }
    return r;
  }
};
// End include: "barrett-reduction.hpp"

template <int id>
struct ArbitraryModIntBase {
  int x;

  ArbitraryModIntBase() : x(0) {}

  ArbitraryModIntBase(int64_t y) {
    int z = y % get_mod();
    if (z < 0) z += get_mod();
    x = z;
  }

  ArbitraryModIntBase &operator+=(const ArbitraryModIntBase &p) {
    if ((x += p.x) >= get_mod()) x -= get_mod();
    return *this;
  }

  ArbitraryModIntBase &operator-=(const ArbitraryModIntBase &p) {
    if ((x += get_mod() - p.x) >= get_mod()) x -= get_mod();
    return *this;
  }

  ArbitraryModIntBase &operator*=(const ArbitraryModIntBase &p) {
    x = rem((unsigned long long)x * p.x);
    return *this;
  }

  ArbitraryModIntBase &operator/=(const ArbitraryModIntBase &p) {
    *this *= p.inverse();
    return *this;
  }

  ArbitraryModIntBase operator-() const { return ArbitraryModIntBase(-x); }
  ArbitraryModIntBase operator+() const { return *this; }

  ArbitraryModIntBase operator+(const ArbitraryModIntBase &p) const {
    return ArbitraryModIntBase(*this) += p;
  }

  ArbitraryModIntBase operator-(const ArbitraryModIntBase &p) const {
    return ArbitraryModIntBase(*this) -= p;
  }

  ArbitraryModIntBase operator*(const ArbitraryModIntBase &p) const {
    return ArbitraryModIntBase(*this) *= p;
  }

  ArbitraryModIntBase operator/(const ArbitraryModIntBase &p) const {
    return ArbitraryModIntBase(*this) /= p;
  }

  bool operator==(const ArbitraryModIntBase &p) const { return x == p.x; }

  bool operator!=(const ArbitraryModIntBase &p) const { return x != p.x; }

  ArbitraryModIntBase inverse() const {
    int a = x, b = get_mod(), u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ArbitraryModIntBase(u);
  }

  ArbitraryModIntBase pow(int64_t n) const {
    ArbitraryModIntBase ret(1), mul(x);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const ArbitraryModIntBase &p) {
    return os << p.x;
  }

  friend istream &operator>>(istream &is, ArbitraryModIntBase &a) {
    int64_t t;
    is >> t;
    a = ArbitraryModIntBase(t);
    return (is);
  }

  int get() const { return x; }

  inline unsigned int rem(unsigned long long p) { return barrett().rem(p); }

  static inline Barrett &barrett() {
    static Barrett b;
    return b;
  }

  static inline int &get_mod() {
    static int mod = 0;
    return mod;
  }

  static void set_mod(int md) {
    assert(0 < md && md <= (1LL << 30) - 1);
    get_mod() = md;
    barrett() = Barrett(md);
  }
};

using ArbitraryModInt = ArbitraryModIntBase<-1>;

/**
 * @brief modint (2^{30} 未満の任意 mod 用)
 */
// End include: "../../modint/arbitrary-modint.hpp"
using namespace yamada;
using atcoder::pow_mod;

ll query(ll i,ll j){
	out("?",i,j);
	cout.flush();
	inl(b);
	return b;
}

void answer(ll ans){
	out("!",ans);
	cout.flush();
}

ll cal(ll N,ll M,ll L,ll B){
	if(N==0){
		if(M>0)return 0;
		if(L>0)return 1%B;
		return 0;
	}
	auto pc=factor_count(B);
	if(pc.empty())return 0;
	if(pc.size()>=2){
		vl mod,rem;
		each2(p,c,pc){
			/* out(p,c); cout.flush(); */
			ll m=Power(p,c);
			ll r=cal(N,M,L,m);
			mod.eb(m);
			rem.eb(r);
		}
		return internal::crt(rem,mod).fi;
	}

	auto[p,c]=(*pc.begin());
	ll mod=Power(p,c);
	if(N%p==0){
		ll cnt=0;
		{
			ll n=N;
			while(n%p==0){ n/=p; ++cnt; }
		}
		ll t=1;
		rep(L){
			t*=M;
			if(cnt*t>=c)break;
		}
		assert(t<=1e9);

		if(cnt*t>=c)return 0;
		/* out("A"); */
		return pow_mod(N,t,B);
	}
	ll t=pow_mod(M,L,B*(p-1)/p);
	return pow_mod(N,t,B);
}

ll nai(ll N,ll M,ll L,ll B){
	ll ML=1;
	rep(L)ML*=M;
	ll ans=1;
	rep(ML)ans*=N;
	return ans%B;
}

void yamada::solve()
{
	inl(N,M,L);
	auto check=[&](ll x)->bool{
		return query(x,1)==x;
	};

	ll B=bisect<ll>(0,1e9,check);
	answer(cal(N,M,L,B));

	/* inl(N,M,L,B); */
	/* out(cal(N,M,L,B),nai(N,M,L,B)); cout.flush(); */
	/* assert(cal(N,M,L,B)==nai(N,M,L,B)); */
}