#include <type_traits>
#include <fstream>
#include <random>
#include <stack>
#include <bitset>
#include <chrono>
#include <optional>
#include <iomanip>
#include <unordered_map>
#include <array>
#include <deque>
#include <string>
#include <utility>
#include <functional>
#include <map>
#include <set>
#include <ostream>
#include <iostream>
#include <numeric>
#include <istream>
#include <queue>
#include <unordered_set>
#include <cassert>
#include <iterator>
#include <algorithm>
#include <vector>

#ifndef KK2_TEMPLATE_PROCON_HPP
#define KK2_TEMPLATE_PROCON_HPP 1


#ifndef KK2_TEMPLATE_CONSTANT_HPP
#define KK2_TEMPLATE_CONSTANT_HPP 1

#ifndef KK2_TEMPLATE_TYPE_ALIAS_HPP
#define KK2_TEMPLATE_TYPE_ALIAS_HPP 1


using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

using pi = std::pair<int, int>;
using pl = std::pair<i64, i64>;
using pil = std::pair<int, i64>;
using pli = std::pair<i64, int>;

template <class T> using vc = std::vector<T>;
template <class T> using vvc = std::vector<vc<T>>;
template <class T> using vvvc = std::vector<vvc<T>>;
template <class T> using vvvvc = std::vector<vvvc<T>>;

template <class T> using pq = std::priority_queue<T>;
template <class T> using pqi = std::priority_queue<T, std::vector<T>, std::greater<T>>;

#endif // KK2_TEMPLATE_TYPE_ALIAS_HPP

template <class T> constexpr T infty = 0;
template <> constexpr int infty<int> = (1 << 30) - 123;
template <> constexpr i64 infty<i64> = (1ll << 62) - (1ll << 31);
template <> constexpr i128 infty<i128> = (i128(1) << 126) - (i128(1) << 63);
template <> constexpr u32 infty<u32> = infty<int>;
template <> constexpr u64 infty<u64> = infty<i64>;
template <> constexpr u128 infty<u128> = infty<i128>;
template <> constexpr double infty<double> = infty<i64>;
template <> constexpr long double infty<long double> = infty<i64>;

constexpr int mod = 998244353;
constexpr int modu = 1e9 + 7;
constexpr long double PI = 3.14159265358979323846;

#endif // KK2_TEMPLATE_CONSTANT_HPP
#ifndef KK2_TEMPLATE_FUNCTION_UTIL_HPP
#define KK2_TEMPLATE_FUNCTION_UTIL_HPP 1


namespace kk2 {

template <class T, class... Sizes> auto make_vector(int first, Sizes... sizes) {
    if constexpr (sizeof...(sizes) == 0) {
        return std::vector<T>(first);
    } else {
        return std::vector<decltype(make_vector<T>(sizes...))>(first, make_vector<T>(sizes...));
    }
}

template <class T, class U> void fill_all(std::vector<T> &v, const U &x) {
    std::fill(std::begin(v), std::end(v), T(x));
}

template <class T, class U> void fill_all(std::vector<std::vector<T>> &v, const U &x) {
    for (auto &u : v) fill_all(u, x);
}

template <class C> int mysize(const C &c) {
    return c.size();
}

// T: comutative monoid
template <class T, class U> U all_sum(const std::vector<T> &v, U init, U = U()) {
    U res = init;
    for (const auto &x : v) res += x;
    return res;
}

template <class T, class U> U all_sum(const std::vector<std::vector<T>> &v, U init, U unit = U()) {
    U res = init;
    for (const auto &u : v) res += all_sum(u, unit, unit);
    return res;
}

// T: commutative monoid, F: (U, T) -> U
template <class T, class U, class F> U all_prod(const std::vector<T> &v, U init, const F &f, U = U()) {
    U res = init;
    for (const auto &x : v) res = f(res, x);
    return res;
}

template <class T, class U, class F>
U all_prod(const std::vector<std::vector<T>> &v, U init, const F &f, U unit) {
    U res = init;
    for (const auto &u : v) res = f(res, all_prod(u, unit, f, unit));
    return res;
}

template <class T> T all_min(const std::vector<T> &v) {
    if (v.empty()) return T();
    T res = v[0];
    for (const auto &x : v) res = res > x ? x : res;
    return res;
}

template <class T>
T all_min(const std::vector<std::vector<T>> &v) {
    T res{};
    bool first = true;
    for (const auto &u : v) {
        if (u.empty()) continue;
        if (first) {
            res = all_min(u);
            first = false;
        } else {
            T tmp = all_min(u);
            res = res > tmp ? tmp : res;
        }
    }
    return res;
}

template <class T> T all_max(const std::vector<T> &v) {
    if (v.empty()) return T();
    T res = v[0];
    for (const auto &x : v) res = res < x ? x : res;
    return res;
}

template <class T>
T all_max(const std::vector<std::vector<T>> &v) {
    T res{};
    bool first = true;
    for (const auto &u : v) {
        if (u.empty()) continue;
        if (first) {
            res = all_max(u);
            first = false;
        } else {
            T tmp = all_max(u);
            res = res < tmp ? tmp : res;
        }
    }
    return res;
}

} // namespace kk2

#endif // KK2_TEMPLATE_FUNCTION_UTIL_HPP
#ifndef KK2_TEMPLATE_IO_UTIL_HPP
#define KK2_TEMPLATE_IO_UTIL_HPP 1


#ifndef KK2_TYPE_TRAITS_TYPE_TRAITS_HPP
#define KK2_TYPE_TRAITS_TYPE_TRAITS_HPP 1



namespace kk2 {

#ifndef _MSC_VER

template <typename T>
using is_signed_int128 = typename std::conditional<std::is_same<T, __int128_t>::value
                                                       or std::is_same<T, __int128>::value,
                                                   std::true_type,
                                                   std::false_type>::type;

template <typename T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value
                                  or std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <typename T>
using is_integral =
    typename std::conditional<std::is_integral<T>::value or is_signed_int128<T>::value
                                  or is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <typename T>
using is_signed = typename std::conditional<std::is_signed<T>::value or is_signed_int128<T>::value,
                                            std::true_type,
                                            std::false_type>::type;

template <typename T>
using is_unsigned =
    typename std::conditional<std::is_unsigned<T>::value or is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <typename T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t, unsigned __int128>;

template <typename T>
using to_unsigned =
    typename std::conditional<is_signed_int128<T>::value,
                              make_unsigned_int128<T>,
                              typename std::conditional<std::is_signed<T>::value,
                                                        std::make_unsigned<T>,
                                                        std::common_type<T>>::type>::type;

#else

template <typename T> using is_integral = std::enable_if_t<std::is_integral<T>::value>;
template <typename T> using is_signed = std::enable_if_t<std::is_signed<T>::value>;
template <typename T> using is_unsigned = std::enable_if_t<std::is_unsigned<T>::value>;
template <typename T> using to_unsigned = std::make_unsigned<T>;

#endif // _MSC_VER

template <typename T> using is_integral_t = std::enable_if_t<is_integral<T>::value>;
template <typename T> using is_signed_t = std::enable_if_t<is_signed<T>::value>;
template <typename T> using is_unsigned_t = std::enable_if_t<is_unsigned<T>::value>;


template <typename T>
using is_function_pointer =
    typename std::conditional<std::is_pointer_v<T> && std::is_function_v<std::remove_pointer_t<T>>,
                              std::true_type,
                              std::false_type>::type;

template <typename T, std::enable_if_t<is_function_pointer<T>::value> * = nullptr>
struct is_two_args_function_pointer : std::false_type {};

template <typename R, typename T1, typename T2>
struct is_two_args_function_pointer<R (*)(T1, T2)> : std::true_type {};

template <typename T>
using is_two_args_function_pointer_t = std::enable_if_t<is_two_args_function_pointer<T>::value>;

namespace type_traits {

struct istream_tag {};

struct ostream_tag {};

} // namespace type_traits

template <typename T>
using is_standard_istream = typename std::conditional<std::is_same<T, std::istream>::value
                                                          || std::is_same<T, std::ifstream>::value,
                                                      std::true_type,
                                                      std::false_type>::type;
template <typename T>
using is_standard_ostream = typename std::conditional<std::is_same<T, std::ostream>::value
                                                          || std::is_same<T, std::ofstream>::value,
                                                      std::true_type,
                                                      std::false_type>::type;
template <typename T> using is_user_defined_istream = std::is_base_of<type_traits::istream_tag, T>;
template <typename T> using is_user_defined_ostream = std::is_base_of<type_traits::ostream_tag, T>;

template <typename T>
using is_istream =
    typename std::conditional<is_standard_istream<T>::value || is_user_defined_istream<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <typename T>
using is_ostream =
    typename std::conditional<is_standard_ostream<T>::value || is_user_defined_ostream<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <typename T> using is_istream_t = std::enable_if_t<is_istream<T>::value>;
template <typename T> using is_ostream_t = std::enable_if_t<is_ostream<T>::value>;


} // namespace kk2

#endif // KK2_TYPE_TRAITS_TYPE_TRAITS_HPP

// なんかoj verifyはプロトタイプ宣言が落ちる

namespace impl {

struct read {
    template <class IStream, class T>
    inline static void all_read(IStream &is, T &x) {
        is >> x;
    }

    template <class IStream, class T, class U>
    inline static void all_read(IStream &is, std::pair<T, U> &p) {
        all_read(is, p.first);
        all_read(is, p.second);
    }

    template <class IStream, class T>
    inline static void all_read(IStream &is, std::vector<T> &v) {
        for (T &x : v) all_read(is, x);
    }

    template <class IStream, class T, size_t F>
    inline static void all_read(IStream &is, std::array<T, F> &a) {
        for (T &x : a) all_read(is, x);
    }
};

struct write {
    template <class OStream, class T>
    inline static void all_write(OStream &os, const T &x) {
        os << x;
    }

    template <class OStream, class T, class U>
    inline static void all_write(OStream &os, const std::pair<T, U> &p) {
        all_write(os, p.first);
        all_write(os, ' ');
        all_write(os, p.second);
    }

    template <class OStream, class T>
    inline static void all_write(OStream &os, const std::vector<T> &v) {
        for (int i = 0; i < (int)v.size(); ++i) {
            if (i) all_write(os, ' ');
            all_write(os, v[i]);
        }
    }

    template <class OStream, class T, size_t F>
    inline static void all_write(OStream &os, const std::array<T, F> &a) {
        for (int i = 0; i < (int)F; ++i) {
            if (i) all_write(os, ' ');
            all_write(os, a[i]);
        }
    }
};

} // namespace impl

template <class IStream, class T, class U, kk2::is_istream_t<IStream> * = nullptr>
IStream &operator>>(IStream &is, std::pair<T, U> &p) {
    impl::read::all_read(is, p);
    return is;
}

template <class IStream, class T, kk2::is_istream_t<IStream> * = nullptr>
IStream &operator>>(IStream &is, std::vector<T> &v) {
    impl::read::all_read(is, v);
    return is;
}

template <class IStream, class T, size_t F, kk2::is_istream_t<IStream> * = nullptr>
IStream &operator>>(IStream &is, std::array<T, F> &a) {
    impl::read::all_read(is, a);
    return is;
}

template <class OStream, class T, class U, kk2::is_ostream_t<OStream> * = nullptr>
OStream &operator<<(OStream &os, const std::pair<T, U> &p) {
    impl::write::all_write(os, p);
    return os;
}

template <class OStream, class T, kk2::is_ostream_t<OStream> * = nullptr>
OStream &operator<<(OStream &os, const std::vector<T> &v) {
    impl::write::all_write(os, v);
    return os;
}

template <class OStream, class T, size_t F, kk2::is_ostream_t<OStream> * = nullptr>
OStream &operator<<(OStream &os, const std::array<T, F> &a) {
    impl::write::all_write(os, a);
    return os;
}

#endif // KK2_TEMPLATE_IO_UTIL_HPP
#ifndef KK2_TEMPLATE_MACROS_HPP
#define KK2_TEMPLATE_MACROS_HPP 1

#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 repi2(i, a) for (long long i = (a) - 1; i >= 0; --i)
#define repi3(i, a, b) for (long long i = (a) - 1; i >= (long long)(b); --i)
#define overload3(a, b, c, d, ...) d
#define rep(...) overload3(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi(...) overload3(__VA_ARGS__, repi3, repi2, rep1)(__VA_ARGS__)

#define fi first
#define se second
#define all(p) p.begin(), p.end()

#endif // KK2_TEMPLATE_MACROS_HPP

struct FastIOSetUp {
    FastIOSetUp() {
        std::ios::sync_with_stdio(false);
        std::cin.tie(nullptr);
    }
} fast_io_set_up;

auto &kin = std::cin;
auto &kout = std::cout;
auto (*kendl)(std::ostream &) = std::endl<char, std::char_traits<char>>;

void Yes(bool b = 1) {
    kout << (b ? "Yes\n" : "No\n");
}

void No(bool b = 1) {
    kout << (b ? "No\n" : "Yes\n");
}

void YES(bool b = 1) {
    kout << (b ? "YES\n" : "NO\n");
}

void NO(bool b = 1) {
    kout << (b ? "NO\n" : "YES\n");
}

void yes(bool b = 1) {
    kout << (b ? "yes\n" : "no\n");
}

void no(bool b = 1) {
    kout << (b ? "no\n" : "yes\n");
}

template <class T, class S> inline bool chmax(T &a, const S &b) {
    return (a < b ? a = b, 1 : 0);
}

template <class T, class S> inline bool chmin(T &a, const S &b) {
    return (a > b ? a = b, 1 : 0);
}

std::istream &operator>>(std::istream &is, u128 &x) {
    std::string s;
    is >> s;
    x = 0;
    for (char c : s) {
        assert('0' <= c && c <= '9');
        x = x * 10 + c - '0';
    }
    return is;
}

std::istream &operator>>(std::istream &is, i128 &x) {
    std::string s;
    is >> s;
    bool neg = s[0] == '-';
    x = 0;
    for (int i = neg; i < (int)s.size(); i++) {
        assert('0' <= s[i] && s[i] <= '9');
        x = x * 10 + s[i] - '0';
    }
    if (neg) x = -x;
    return is;
}

std::ostream &operator<<(std::ostream &os, u128 x) {
    if (x == 0) return os << '0';
    std::string s;
    while (x) {
        s.push_back('0' + x % 10);
        x /= 10;
    }
    std::reverse(s.begin(), s.end());
    return os << s;
}

std::ostream &operator<<(std::ostream &os, i128 x) {
    if (x == 0) return os << '0';
    if (x < 0) {
        os << '-';
        x = -x;
    }
    std::string s;
    while (x) {
        s.push_back('0' + x % 10);
        x /= 10;
    }
    std::reverse(s.begin(), s.end());
    return os << s;
}

#endif // KK2_TEMPLATE_PROCON_HPP
// #include <kk2/template/debug.hpp>
#ifndef KK2_MODINT_MODINT_HPP
#define KK2_MODINT_MODINT_HPP 1



namespace kk2 {

template <int p> struct ModInt {
    using mint = ModInt;

  public:
    static int Mod;

    constexpr static unsigned int getmod() {
        if (p > 0) return p;
        else return Mod;
    }

    static void setmod(int Mod_) {
        assert(1 <= Mod_);
        Mod = Mod_;
    }

    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    constexpr ModInt() : _v(0) {}

    template <class T, is_integral_t<T> * = nullptr> constexpr ModInt(T v) {
        if constexpr (is_signed<T>::value) {
            v = v % (long long)(getmod());
            if (v < 0) v += getmod();
            _v = v;
        } else if constexpr (is_unsigned<T>::value) {
            _v = v %= getmod();
        } else {
            ModInt();
        }
    }

    unsigned int val() const { return _v; }

    mint &operator++() {
        _v++;
        if (_v == getmod()) _v = 0;
        return *this;
    }

    mint &operator--() {
        if (_v == 0) _v = getmod();
        _v--;
        return *this;
    }

    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }

    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint &operator+=(const mint &rhs) {
        _v += rhs._v;
        if (_v >= getmod()) _v -= getmod();
        return *this;
    }

    mint &operator-=(const mint &rhs) {
        _v += getmod() - rhs._v;
        if (_v >= getmod()) _v -= getmod();
        return *this;
    }

    mint &operator*=(const mint &rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        z %= getmod();
        _v = z;
        return *this;
    }

    mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }

    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }

    mint inv() const {
        long long s = getmod(), t = _v;
        long long m0 = 0, m1 = 1;

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

            std::swap(s, t);
            std::swap(m0, m1);
        }
        if (m0 < 0) m0 += getmod() / s;
        return m0;
    }

    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }

    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }

    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }

    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }

    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }

    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }

    template <class OStream, is_ostream_t<OStream> * = nullptr>
    friend OStream &operator<<(OStream &os, const mint &mint_) {
        os << mint_._v;
        return os;
    }

    template <class IStream, is_istream_t<IStream> * = nullptr>
    friend IStream &operator>>(IStream &is, mint &mint_) {
        long long x;
        is >> x;
        mint_ = mint(x);
        return is;
    }

  private:
    unsigned int _v;
};

template <int p> int ModInt<p>::Mod = 998244353;


using mint998 = ModInt<998244353>;
using mint107 = ModInt<1000000007>;

} // namespace kk2

#endif // KK2_MODINT_MODINT_HPP
#ifndef KK2_RANDOM_GEN_HPP
#define KK2_RANDOM_GEN_HPP 1


#ifndef KK2_RANDOM_SEED_HPP
#define KK2_RANDOM_SEED_HPP 1


namespace kk2 {

namespace random {

using u64 = unsigned long long;

inline u64 non_deterministic_seed() {
    u64 seed = std::chrono::duration_cast<std::chrono::nanoseconds>(
                   std::chrono::high_resolution_clock::now().time_since_epoch())
                   .count();
    seed ^= reinterpret_cast<u64>(&seed);
    seed ^= seed << 5;
    seed ^= seed >> 41;
    seed ^= seed << 20;
    return seed;
}

inline u64 deterministic_seed() {
    return 5801799128519729247ull;
}

inline u64 seed() {
#if defined(KK2_RANDOM_DETERMINISTIC)
    return deterministic_seed();
#else
    return non_deterministic_seed();
#endif
}

} // namespace random

} // namespace kk2

#endif // KK2_RANDOM_SEED_HPP

namespace kk2 {

namespace random {

using i64 = long long;
using u64 = unsigned long long;

inline u64 rng() {
    static std::mt19937_64 mt(kk2::random::seed());
    return mt();
}

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

// [l, r)
template <class T> std::vector<T> random_vector(int n, T l, T r) {
    std::vector<T> res(n);
    for (int i = 0; i < n; i++) res[i] = rng(l, r);
    return res;
}

// [l, r)
std::vector<i64> distinct_rng(i64 l, i64 r, i64 n) {
    assert(l < r and n <= r - l);
    std::unordered_set<i64> st;
    for (i64 i = n; i; --i) {
        i64 m = rng(l, r + 1 - i);
        if (st.find(m) != st.end()) m = r - i;
        st.insert(m);
    }
    std::vector<i64> res(st.begin(), st.end());
    std::sort(res.begin(), res.end());
    return res;
}

template <class Iter> void shuffle(Iter first, Iter last) {
    if (first == last) return;
    int len = 1;
    for (auto it = first + 1; it != last; ++it) {
        len++;
        int j = rng(0, len);
        if (j != len - 1) std::iter_swap(first + j, it);
    }
}

template <class T> std::vector<T> perm(int n) {
    std::vector<T> res(n);
    std::iota(res.begin(), res.end(), T(0));
    shuffle(res.begin(), res.end());
    return res;
}

template <class T> std::vector<T> choices(int l, int r, int k) {
    assert(l < r and k <= r - l);
    std::vector<T> res(r - l);
    std::iota(res.begin(), res.end(), T(l));
    shuffle(res.begin(), res.end());
    res.resize(k);
    return res;
}

} // namespace random

} // namespace kk2

#endif // KK2_RANDOM_GEN_HPP
#ifndef KK2_MATH_IS_PRIME_HPP
#define KK2_MATH_IS_PRIME_HPP 1


#ifndef KK2_MATH_MOD_POW_MOD_HPP
#define KK2_MATH_MOD_POW_MOD_HPP 1



namespace kk2 {

template <class S, class T, class U> constexpr S pow_mod(T x, U n, T m) {
    assert(n >= 0);
    if (m == 1) return S(0);
    S _m = m, r = 1;
    S y = x % _m;
    if (y < 0) y += _m;
    while (n) {
        if (n & 1) r = (r * y) % _m;
        if (n >>= 1) y = (y * y) % _m;
    }
    return r;
}

} // namespace kk2

#endif // KK2_MATH_MOD_POW_MOD_HPP
#ifndef KK2_MODINT_MONT_ARB_HPP
#define KK2_MODINT_MONT_ARB_HPP 1



namespace kk2 {

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 setmod(UInt m) {
        assert(m < (UInt(1u) << (bit_length - 2)));
        assert((m & 1) == 1);
        mod = m, n2 = -ULong(m) % m, r = get_r();
    }

    UInt _v;

    ArbitraryLazyMontgomeryModIntBase() : _v(0) {}

    template <class T, is_integral_t<T> * = nullptr>
    ArbitraryLazyMontgomeryModIntBase(const T &b) : _v(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(_v += b._v - 2 * mod) < 0) _v += 2 * mod;
        return *this;
    }

    mint &operator-=(const mint &b) {
        if (Int(_v -= b._v) < 0) _v += 2 * mod;
        return *this;
    }

    mint &operator*=(const mint &b) {
        _v = reduce(ULong(_v) * b._v);
        return *this;
    }

    mint &operator/=(const mint &b) {
        *this *= b.inv();
        return *this;
    }

    friend mint operator+(const mint &a, const mint &b) { return mint(a) += b; }

    friend mint operator-(const mint &a, const mint &b) { return mint(a) -= b; }

    friend mint operator*(const mint &a, const mint &b) { return mint(a) *= b; }

    friend mint operator/(const mint &a, const mint &b) { return mint(a) /= b; }

    bool operator==(const mint &b) const {
        return (_v >= mod ? _v - mod : _v) == (b._v >= mod ? b._v - mod : b._v);
    }

    bool operator!=(const mint &b) const {
        return (_v >= mod ? _v - mod : _v) != (b._v >= mod ? b._v - mod : b._v);
    }

    mint operator-() const { return mint(0) - mint(*this); }

    mint operator+() const { return mint(*this); }

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

    mint inv() const {
        Int s = getmod(), t = val(), m0 = 0, m1 = 1;
        while (t) {
            Int u = s / t;
            std::swap(s -= t * u, t);
            std::swap(m0 -= m1 * u, m1);
        }
        if (m0 < 0) m0 += getmod();
        return mint(m0);
    }

    template <class OStream, is_ostream_t<OStream> * = nullptr>
    friend OStream &operator<<(OStream &os, const mint &x) {
        return os << x.val();
    }

    template <class IStream, is_istream_t<IStream> * = nullptr>
    friend IStream &operator>>(IStream &is, mint &x) {
        Long t;
        is >> t;
        x = mint(t);
        return (is);
    }

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

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

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>;

} // namespace kk2

#endif // KK2_MODINT_MONT_ARB_HPP

namespace kk2 {

namespace number_theory {

template <class T, class U> bool miller_rabin(const T &n, const std::vector<T> &ws) {
    if (n <= 2) return n == 2;
    if (~n & 1) return false;

    T d = n - 1;
    while (~d & 1) d >>= 1;
    U e = 1, rev = n - 1;
    for (T w : ws) {
        if (w % n == 0) continue;
        T t = d;
        U y = pow_mod<T, T, U>(w, t, n);
        while (t != n - 1 and y != e and y != rev) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != rev and ~t & 1) 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 <class mint>
bool miller_rabin_mont(unsigned long long n, const std::vector<unsigned long long> &ws) {
    if (n <= 2) return n == 2;
    if (~n & 1) return false;

    if (mint::getmod() != n) mint::setmod(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 and y != e and y != rev) {
            y *= y;
            t <<= 1;
        }
        if (y != rev and ~t & 1) return false;
    }
    return true;
}

bool is_prime(unsigned long long n) {
    using mint32 = ArbitraryLazyMontgomeryModInt<54305750>;
    using mint64 = ArbitraryLazyMontgomeryModInt64bit<54305750>;

    if (n <= 2) return n == 2;
    if (~n & 1) return false;
    if (n < (1ull << 30)) {
        return miller_rabin_mont<mint32>(n, {2, 7, 61});
    } else if (n < (1ull << 62)) {
        return miller_rabin_mont<mint64>(n, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
    } else {
        return miller_rabin_u64(n);
    }
}

}; // namespace number_theory

using number_theory::is_prime;

}; // namespace kk2

#endif // KK2_MATH_IS_PRIME_HPP
using namespace std;

void solve() {
    /*
    ロリハしたいけど，小さい素数だとフェル小でループする
    素数を乱択したら，hackされないか？
    */

    int k;
    kin >> k;
    vc<pair<string, i64>> a(k);
    kin >> a;
    // using mint = kk2::ArbitraryLazyMontgomeryModInt<-1>;
    using mint = kk2::ModInt<-1>;

    auto is_ok = [&]() -> bool {
        mint base = kk2::random::rng(2, mint::getmod() - 1);
        // base = 2;
        mint ibase = base.inv();
        mint bpw = 1;
        mint r1, r2;
        rep (i, k) {
            auto [s, t] = a[i];
            mint x1 = 0, x2 = 0;
            repi (j, s.size()) {
                x1 = x1 * base + mint(s[j]);
                x2 = x2 * ibase + mint(s[j]);
            }
            if (base.pow(s.size()) == mint(1)) return true;
            if (ibase.pow(s.size()) == mint(1)) return true;
            x1 *= base.pow(t).pow(s.size()) - 1;
            x1 /= base.pow(s.size()) - 1;
            x2 *= ibase.pow(t).pow(s.size()) - 1;
            x2 /= ibase.pow(s.size()) - 1;
            r1 += x1 * bpw, r2 += x2 * bpw.inv();
            bpw *= base.pow(t).pow(s.size());
        }
        // kout << bpw << kendl;    
        return base * r1 == r2 * bpw;
    };

    int t = 10;
    rep (t) {
        int p;
        do {
            p = kk2::random::rng(1e8, 1e9);
        } while (!kk2::is_prime(p));
        mint::setmod(p);
        // kout << mint::getmod() << kendl;
        if (!is_ok()) {
            No();
            return;
        }
    }
    Yes();
}

int main() {
    int t = 1;
    // kin >> t;
    rep (t) solve();

    return 0;
}
// Author: kk2
// converted by https://github.com/kk2a/cpp-bundle
// 2025-04-01 10:24:48