#line 2 "library/KowerKoint/stl-expansion.hpp"
#include <bits/stdc++.h>

template <typename T1, typename T2>
std::istream& operator>>(std::istream& is, std::pair<T1, T2>& p) {
    is >> p.first >> p.second;
    return is;
}
template <typename T, size_t N>
std::istream& operator>>(std::istream& is, std::array<T, N>& a) {
    for (size_t i = 0; i < N; ++i) {
        is >> a[i];
    }
    return is;
}
template <typename T>
std::istream& operator>>(std::istream& is, std::vector<T>& v) {
    for (auto& e : v) is >> e;
    return is;
}
template <typename T1, typename T2>
std::ostream& operator<<(std::ostream& os, const std::pair<T1, T2>& p) {
    os << p.first << " " << p.second;
    return os;
}
template <typename T, size_t N>
std::ostream& operator<<(std::ostream& os, const std::array<T, N>& a) {
    for (size_t i = 0; i < N; ++i) {
        os << a[i] << (i + 1 == a.size() ? "" : " ");
    }
    return os;
}
template <typename T>
std::ostream& operator<<(std::ostream& os, const std::vector<T>& v) {
    for (size_t i = 0; i < v.size(); ++i) {
        os << v[i] << (i + 1 == v.size() ? "" : " ");
    }
    return os;
}
#line 3 "library/KowerKoint/base.hpp"
using namespace std;

#define REP(i, n) for(int i = 0; i < (int)(n); i++)
#define FOR(i, a, b) for(ll i = a; i < (ll)(b); i++)
#define ALL(a) (a).begin(),(a).end()
#define RALL(a) (a).rbegin(),(a).rend()
#define END(...) { print(__VA_ARGS__); return; }

using VI = vector<int>;
using VVI = vector<VI>;
using VVVI = vector<VVI>;
using ll = long long;
using VL = vector<ll>;
using VVL = vector<VL>;
using VVVL = vector<VVL>;
using ull = unsigned long long;
using VUL = vector<ull>;
using VVUL = vector<VUL>;
using VVVUL = vector<VVUL>;
using VD = vector<double>;
using VVD = vector<VD>;
using VVVD = vector<VVD>;
using VS = vector<string>;
using VVS = vector<VS>;
using VVVS = vector<VVS>;
using VC = vector<char>;
using VVC = vector<VC>;
using VVVC = vector<VVC>;
using P = pair<int, int>;
using VP = vector<P>;
using VVP = vector<VP>;
using VVVP = vector<VVP>;
using LP = pair<ll, ll>;
using VLP = vector<LP>;
using VVLP = vector<VLP>;
using VVVLP = vector<VVLP>;

template <typename T>
using PQ = priority_queue<T>;
template <typename T>
using GPQ = priority_queue<T, vector<T>, greater<T>>;

constexpr int INF = 1001001001;
constexpr ll LINF = 1001001001001001001ll;
constexpr int DX[] = {1, 0, -1, 0};
constexpr int DY[] = {0, 1, 0, -1};

void print() { cout << '\n'; }
template<typename T>
void print(const T &t) { cout << t << '\n'; }
template<typename Head, typename... Tail>
void print(const Head &head, const Tail &... tail) {
    cout << head << ' ';
    print(tail...);
}

#ifdef DEBUG
void dbg() { cerr << '\n'; }
template<typename T>
void dbg(const T &t) { cerr << t << '\n'; }
template<typename Head, typename... Tail>
void dbg(const Head &head, const Tail &... tail) {
    cerr << head << ' ';
    dbg(tail...);
}
#else
template<typename... Args>
void dbg(const Args &... args) {}
#endif

template<typename T>
vector<vector<T>> split(typename vector<T>::const_iterator begin, typename vector<T>::const_iterator end, T val) {
    vector<vector<T>> res;
    vector<T> cur;
    for(auto it = begin; it != end; it++) {
        if(*it == val) {
            res.push_back(cur);
            cur.clear();
        } else cur.push_back(*it);
    }
    res.push_back(cur);
    return res;
}

vector<string> split(typename string::const_iterator begin, typename string::const_iterator end, char val) {
    vector<string> res;
    string cur = "";
    for(auto it = begin; it != end; it++) {
        if(*it == val) {
            res.push_back(cur);
            cur.clear();
        } else cur.push_back(*it);
    }
    res.push_back(cur);
    return res;
}

template< typename T1, typename T2 >
inline bool chmax(T1 &a, T2 b) { return a < b && (a = b, true); }

template< typename T1, typename T2 >
inline bool chmin(T1 &a, T2 b) { return a > b && (a = b, true); }

template <typename T>
pair<VI, vector<T>> compress(const vector<T> &a) {
    int n = a.size();
    vector<T> x;
    REP(i, n) x.push_back(a[i]);
    sort(ALL(x)); x.erase(unique(ALL(x)), x.end());
    VI res(n);
    REP(i, n) res[i] = lower_bound(ALL(x), a[i]) - x.begin();
    return make_pair(res, x);
}

template <typename It>
auto rle(It begin, It end) {
    vector<pair<typename It::value_type, int>> res;
    if(begin == end) return res;
    auto pre = *begin;
    int num = 1;
    for(auto it = begin + 1; it != end; it++) {
        if(pre != *it) {
            res.emplace_back(pre, num);
            pre = *it;
            num = 1;
        } else num++;
    }
    res.emplace_back(pre, num);
    return res;
}

template <typename It>
vector<pair<typename It::value_type, int>> rle_sort(It begin, It end) {
    vector<typename It::value_type> cloned(begin, end);
    sort(ALL(cloned));
    auto e = rle(ALL(cloned));
    sort(ALL(e), [](const auto& l, const auto& r) { return l.second < r.second; });
    return e;
}

template <typename T>
pair<vector<T>, vector<T>> factorial(int n) {
    vector<T> res(n+1), rev(n+1);
    res[0] = 1;
    REP(i, n) res[i+1] = res[i] * (i+1);
    rev[n] = 1 / res[n];
    for(int i = n; i > 0; i--) {
        rev[i-1] = rev[i] * i;
    }
    return make_pair(res, rev);
}
#line 3 "library/KowerKoint/integer/extgcd.hpp"

constexpr ll extgcd(ll a, ll b, ll& x, ll& y) {
    x = 1, y = 0;
    ll nx = 0, ny = 1;
    while(b) {
        ll q = a / b;
        ll r = a % b;
        a = b, b = r;
        ll nnx = x - q * nx;
        ll nny = y - q * ny;
        x = nx, nx = nnx;
        y = ny, ny = nny;
    }
    return a;
}
#line 3 "library/KowerKoint/integer/pow-mod.hpp"

constexpr ll inv_mod(ll n, ll m) {
    n %= m;
    if (n < 0) n += m;
    ll x = -1, y = -1;
    if(extgcd(n, m, x, y) != 1) throw logic_error("");
    x %= m;
    if(x < 0) x += m;
    return x;
}

constexpr ll pow_mod(ll a, ll n, ll m) {
    if(n == 0) return 1LL;
    if(n < 0) return inv_mod(pow_mod(a, -n, m), m);
    a %= m;
    if (a < 0) n += m;
    ll res = 1;
    while(n) {
        if(n & 1) {
            res *= a;
            res %= m;
        }
        n >>= 1;
        a *= a;
        a %= m;
    }
    return res;
}
#line 3 "library/KowerKoint/algebra/field.hpp"

template <typename T>
struct SumGroupBase {
    constexpr static bool defzero = false;
    using Coef = nullptr_t;
    using Scalar = nullptr_t;
};
template <typename T>
struct ProdGroupBase {
    constexpr static bool defone = false;
};
template <typename T>
struct RepresentationBase {
    using R = T;
    constexpr static T construct(const R& x) { return x; }
    constexpr static R represent(const T& x) { return x; }
};
template <typename T>
struct CompareBase {
    constexpr static bool eq(const T& x, const T& y) { return x == y; }
    constexpr static bool lt(const T& x, const T& y) { return x < y; }
};
template <typename T>
struct FinitePropertyBase {
    constexpr static bool is_finite = false;
};

template <typename T, typename SumGroup = SumGroupBase<T>, typename ProdGroup = ProdGroupBase<T>, typename Representation = RepresentationBase<T>, typename Compare = CompareBase<T>, typename FiniteProperty = FinitePropertyBase<T>>
struct Field {
    using R = typename Representation::R;
    using Coef = typename SumGroup::Coef;
    using Scalar = typename SumGroup::Scalar;
    T val;
    constexpr static Field zero() {
        return SumGroup::zero;
    }
    constexpr static Field one() {
        return ProdGroup::one;
    }
    constexpr static bool defzero = SumGroup::defzero;
    constexpr static bool defone = ProdGroup::defone;
    constexpr static bool is_finite = FiniteProperty::is_finite;
    constexpr Field() {
        if constexpr(SumGroup::defzero) val = SumGroup::zero;
        else if constexpr(SumGroup::defone) val = ProdGroup::one;
        else val = T();
    }
    constexpr Field(const R& r) : val(Representation::construct(r)) {}
    constexpr R represent() const { return Representation::represent(val); }
    constexpr decltype(auto) operator[](size_t i) const {
        return val[i];
    }
    constexpr static Field premitive_root() {
        return FiniteProperty::premitive_root();
    }
    constexpr static size_t order() {
        return FiniteProperty::order();
    }
    constexpr Field& operator*=(const Field& other) {
        ProdGroup::mulassign(val, other.val);
        return *this;
    }
    constexpr Field operator*(const Field& other) const {
        return Field(*this) *= other;
    }
    constexpr Field inv() const {
        return ProdGroup::inv(val);
    }
    constexpr Field& operator/=(const Field& other) {
        return *this *= other.inv();
    }
    constexpr Field operator/(const Field& other) const {
        return Field(*this) /= other;
    }
    constexpr Field pow(ll n) const {
        if(n < 0) {
            return inv().pow(-n);
        }
        Field res = one();
        Field a = *this;
        while(n > 0) {
            if(n & 1) res *= a;
            a *= a;
            n >>= 1;
        }
        return res;
    }
    constexpr Field operator+() const {
        return *this;
    }
    constexpr Field& operator+=(const Field& other) {
        SumGroup::addassign(val, other.val);
        return *this;
    }
    constexpr Field operator+(const Field& other) const {
        return Field(*this) += other;
    }
    constexpr Field operator-() const {
        return SumGroup::minus(val);
    }
    constexpr Field& operator-=(const Field& other) {
        return *this += -other;
    }
    constexpr Field operator-(const Field& other) const {
        return Field(*this) -= other;
    }
    constexpr Field& operator++() {
        return *this += one();
    }
    Field operator++(int) {
        Field ret = *this;
        ++*this;
        return ret;
    }
    constexpr Field& operator--() {
        return *this -= one();
    }
    Field operator--(int) {
        Field ret = *this;
        --*this;
        return ret;
    }
    constexpr Field& operator*=(const Coef& other) {
        SumGroup::coefassign(val, other);
        return *this;
    }
    constexpr Field operator*(const Coef& other) const {
        return Field(*this) *= other;
    }
    constexpr Scalar dot(const Field& other) const {
        return SumGroup::dot(val, other.val);
    }
    constexpr Scalar norm() const {
        return dot(*this);
    }
    constexpr bool operator==(const Field& other) const {
        return Compare::eq(val, other.val);
    }
    constexpr bool operator!=(const Field& other) const {
        return !(*this == other);
    }
    constexpr bool operator<(const Field& other) const {
        return Compare::lt(represent(), other.represent());
    }
    constexpr bool operator>(const Field& other) const {
        return other < *this;
    }
    constexpr bool operator<=(const Field& other) const {
        return !(*this > other);
    }
    constexpr bool operator>=(const Field& other) const {
        return !(*this < other);
    }
    friend istream& operator>>(istream& is, Field& f) {
        R r; is >> r;
        f = r;
        return is;
    }
    friend ostream& operator<<(ostream& os, const Field& f) {
        return os << f.represent();
    }
};
namespace std {
    template <typename T>
    struct hash<Field<T>> {
        size_t operator()(const Field<T>& f) const {
            return hash<typename Field<T>::R>()(f.represent());
        }
    };
}
template <typename>
struct is_field : false_type {};
template <typename T, typename SumGroup, typename ProdGroup, typename Representation, typename FiniteProperty>
struct is_field<Field<T, SumGroup, ProdGroup, Representation, FiniteProperty>> : true_type {};
template <typename T>
constexpr bool is_field_v = is_field<T>::value;
template <typename T>
constexpr T zero() {
    if constexpr(is_field_v<T>) return T::zero();
    else return 0;
}
template <typename T>
constexpr T one() {
    if constexpr(is_field_v<T>) return T::one();
    else return 1;
}
template <typename T>
constexpr bool is_finite() {
    if constexpr(is_field_v<T>) return T::is_finite;
    else return false;
}
#line 4 "library/KowerKoint/algebra/modint.hpp"

template <ll mod>
struct SumGroupModint : SumGroupBase<ll> {
    static ll& addassign(ll& l, const ll& r) {
        ll ret;
        if(__builtin_add_overflow(l, r, &ret)) {
            l = l % mod + r % mod;
        } else {
            l = ret;
        }
        return l;
    }
    constexpr static bool defzero = true;
    constexpr static ll zero = 0;
    constexpr static ll minus(const ll& x) {
        return -x;
    }
};
template <ll mod>
struct ProdGroupModint : ProdGroupBase<ll> {
    constexpr static bool defmul = true;
    static ll& mulassign(ll& l, const ll& r) {
        ll ret;
        if(__builtin_mul_overflow(l, r, &ret)) {
            l = (l % mod) * (r % mod);
        } else {
            l = ret;
        }
        return l;
    }
    constexpr static bool defone = true;
    constexpr static ll one = 1;
    constexpr static bool definv = true;
    constexpr static ll inv(const ll& x) {
        return inv_mod(x, mod);
    }
};
template <ll mod>
struct RepresentationModint : RepresentationBase<ll> {
    using R = ll;
    constexpr static ll construct(const R& x) { return x % mod; }
    constexpr static R represent(const ll& x) {
        ll ret = x % mod;
        if(ret < 0) ret += mod;
        return ret;
    }
};
template <ll mod>
struct CompareModint : CompareBase<ll> {
    constexpr static bool lt(const ll& l, const ll& r) {
        return RepresentationModint<mod>::represent(l) < RepresentationModint<mod>::represent(r);
    }
    constexpr static bool eq(const ll& l, const ll& r) {
        return RepresentationModint<mod>::represent(l) == RepresentationModint<mod>::represent(r);
    }
};
template <ll mod>
struct FinitePropertyModint : FinitePropertyBase<ll> {
    constexpr static bool is_finite = true;
    constexpr static ll premitive_root() {
        static_assert(mod == 998244353);
        return 3;
    }
    constexpr static size_t order() {
        return mod - 1;
    }
};

template <ll mod>
using Modint = Field<ll, SumGroupModint<mod>, ProdGroupModint<mod>, RepresentationModint<mod>, CompareModint<mod>, FinitePropertyModint<mod>>;

using MI3 = Modint<998244353>;
using V3 = vector<MI3>;
using VV3 = vector<V3>;
using VVV3 = vector<VV3>;
using MI7 = Modint<1000000007>;
using V7 = vector<MI7>;
using VV7 = vector<V7>;
using VVV7 = vector<VV7>;
#line 3 "library/KowerKoint/counting/counting.hpp"

template <typename T>
struct Counting {
    vector<T> fact, ifact;

    Counting() {}
    Counting(ll n) {
        assert(n >= 0);
        expand(n);
    }

    void expand(ll n) {
        assert(n >= 0);
        ll sz = (ll)fact.size();
        if(sz > n) return;
        fact.resize(n+1);
        ifact.resize(n+1);
        fact[0] = 1;
        FOR(i, max(1LL, sz), n+1) fact[i] = fact[i-1] * i;
        ifact[n] = fact[n].inv();
        for(ll i = n-1; i >= sz; i--) ifact[i] = ifact[i+1] * (i+1);
    }

    T p(ll n, ll r) {
        if(n < r) return 0;
        assert(r >= 0);
        expand(n);
        return fact[n] * ifact[n-r];
    }

    T c(ll n, ll r) {
        if(n < r) return 0;
        assert(r >= 0);
        expand(n);
        return fact[n] * ifact[r] * ifact[n-r];
    }

    T h(ll n, ll r) {
        assert(n >= 0);
        assert(r >= 0);
        return c(n+r-1, r);
    }

    T stirling(ll n, ll k) {
        if(n < k) return 0;
        assert(k >= 0);
        if(n == 0) return 1;
        T res = 0;
        T sign = k%2? -1 : 1;
        expand(k);
        REP(i, k+1) {
            res += sign * ifact[i] * ifact[k-i] * T(i).pow(n);
            sign *= -1;
        }
        return res;
    }

    vector<vector<T>> stirling_table(ll n, ll k) {
        assert(n >= 0 && k >= 0);
        vector<vector<T>> res(n+1, vector<T>(k+1));
        res[0][0] = 1;
        FOR(i, 1, n+1) FOR(j, 1, k+1) {
            res[i][j] = res[i-1][j-1] + j * res[i-1][j];
        }
        return res;
    }

    T bell(ll n, ll k) {
        assert(n >= 0 && k >= 0);
        expand(k);
        vector<T> tmp(k+1);
        T sign = 1;
        tmp[0] = 1;
        FOR(i, 1, k+1) {
            sign *= -1;
            tmp[i] = tmp[i-1] + sign * ifact[i];
        }
        T res = 0;
        REP(i, k+1) {
            res += T(i).pow(n) * ifact[i] * tmp[k-i];
        }
        return res;
    }

    vector<vector<T>> partition_table(ll n, ll k) {
        assert(n >= 0 && k >= 0);
        vector<vector<T>> res(n+1, vector<T>(k+1));
        REP(i, k+1) res[0][i] = 1;
        FOR(i, 1, n+1) FOR(j, 1, k+1) {
            res[i][j] = res[i][j-1] + (i<j? 0 : res[i-j][j]);
        }
        return res;
    }
};
#line 3 "library/KowerKoint/structure/union-find.hpp"

template<typename GroupValue_, typename VertexValue_, typename VertexValueCoef_>
struct UnionFindData {
    GroupValue_ group_value;
    bool vertex_value_determined;
    VertexValue_ vertex_value;
    VertexValueCoef_ a;
    VertexValue_ b;
};
template<>
struct UnionFindData<nullptr_t, nullptr_t, nullptr_t> {
};
template<typename VertexValue_>
struct UnionFindData<nullptr_t, VertexValue_, nullptr_t> {
    bool vertex_value_determined;
    VertexValue_ vertex_value;
    VertexValue_ b;
};
template<typename VertexValue_, typename VertexValueCoef_>
struct UnionFindData<nullptr_t, VertexValue_, VertexValueCoef_> {
    bool vertex_value_determined;
    VertexValue_ vertex_value;
    VertexValueCoef_ a;
    VertexValue_ b;
};
template <typename GroupValue_>
struct UnionFindData<GroupValue_, nullptr_t, nullptr_t> {
    GroupValue_ group_value;
};
template<typename GroupValue_, typename VertexValue_>
struct UnionFindData<GroupValue_, VertexValue_, nullptr_t> {
    GroupValue_ group_value;
    bool vertex_value_determined;
    VertexValue_ vertex_value;
    VertexValue_ b;
};

template <typename GroupValue=nullptr_t, typename MergeGroupValue=nullptr_t, typename VertexValue=nullptr_t, typename VertexValueCoef = nullptr_t>
class UnionFind {
    int _n;
    VI _par; //size if value is negative
    vector<UnionFindData<GroupValue, VertexValue, VertexValueCoef>> _data;
    MergeGroupValue _merge_group_value;

public:
    UnionFind(int n=0) : _n(n), _par(n, -1), _data(n) {
        for(auto &d : _data) {
            if constexpr(!is_same_v<VertexValue, nullptr_t>) {
                d.vertex_value_determined = false;
                d.b = 0;
                if constexpr(!is_same_v<VertexValueCoef, nullptr_t>) {
                    d.a = 1;
                }
            }
        }
    }
    int root(int x) {
        if (_par[x] < 0) return x;
        int r = root(_par[x]);
        if constexpr (!is_same_v<VertexValue, nullptr_t>) {
            if constexpr (is_same_v<VertexValueCoef, nullptr_t>) {
                _data[x].b += _data[_par[x]].b;
            } else {
                _data[x].b =_data[x].a * _data[_par[x]].b + _data[x].b;
                _data[x].a *= _data[_par[x]].a;
            }
        }
        return _par[x] = r;
    }
    GroupValue get_group_value(int x) {
        return _data[root(x)].group_value;
    }
    void set_group_value(int x, GroupValue v) {
        _data[root(x)].group_value = v;
    }
    pair<bool, VertexValue> get_vertex_value(int x) {
        int r = root(x);
        if(!_data[r].vertex_value_determined) return {false, {}};
        if constexpr (is_same_v<VertexValueCoef, nullptr_t>) {
            return {true, _data[r].vertex_value + _data[x].b};
        } else {
            return {true, _data[x].a * _data[r].vertex_value + _data[x].b};
        }
    }
    bool set_vertex_value(int x, VertexValue v) {
        auto [determined, old] = get_vertex_value(x);
        if(determined) return old == v;
        int r = root(x);
        _data[r].vertex_value_determined = true;
        _data[r].vertex_value = (v - _data[x].b) / _data[x].a;
        return true;
    }
    int size(int x) {
        return -_par[root(x)];
    }
    bool merge(int x, int y) {
        static_assert(is_same_v<VertexValue, nullptr_t>);
        x = root(x);
        y = root(y);
        if (x == y) return false;
        if (_par[x] > _par[y]) swap(x, y);
        _par[x] += _par[y];
        _par[y] = x;
        if constexpr (!is_same_v<GroupValue, nullptr_t>) {
            _data[x].group_value = _merge_group_value(_data[x].group_value, _data[y].group_value);
        }
        return true;
    }
    pair<bool, bool> merge(int x, int y, VertexValue b) {
        static_assert(!is_same_v<VertexValue, nullptr_t>);
        static_assert(is_same_v<VertexValueCoef, nullptr_t>);
        int rx = root(x);
        int ry = root(y);
        if (rx == ry) {
            if(_data[x].b + b != _data[y].b) return {false, false};
            return {true, false};
        }
        b += _data[x].b;
        b -= _data[y].b;
        x = rx, y = ry;
        if (_par[x] > _par[y]) {
            swap(x, y);
            b = -b;
        }
        if(_data[y].vertex_value_determined) {
            if(_data[x].vertex_value_determined) {
                if(_data[x].vertex_value + b != _data[y].vertex_value) return {false, false};
            } else {
                _data[x].vertex_value_determined = true;
                _data[x].vertex_value = _data[y].vertex_value - b;
            }
        }
        _par[x] += _par[y];
        _par[y] = x;
        if constexpr (!is_same_v<GroupValue, nullptr_t>) {
            _data[x].group_value = _merge_group_value(_data[x].group_value, _data[y].group_value);
        }
        _data[y].b = b;
        return {true, true};
    }
    pair<bool, bool> merge(int x, int y, VertexValueCoef a, VertexValue b) {
        static_assert(!is_same_v<VertexValue, nullptr_t>);
        static_assert(!is_same_v<VertexValueCoef, nullptr_t>);
        int rx = root(x);
        int ry = root(y);
        if (rx == ry) {
            VertexValueCoef a1 = a * _data[x].a - _data[y].a;
            VertexValue b1 = a * _data[x].b + b - _data[y].b;
            if(a1 == 0) {
                if(b1 != 0) return {false, false};
            } else if(_data[rx].vertex_value_determined) {
                if(a1 * _data[rx].vertex_value + b1 != 0) return {false, false};
            } else {
                _data[rx].vertex_value_determined = true;
                _data[rx].vertex_value = -b1 / a1;
            }
            return {true, false};
        }
        b = a * _data[x].b + b;
        a *= _data[x].a;
        b = (b - _data[y].b) / _data[y].a;
        a /= _data[y].a;
        x = rx, y = ry;
        if (_par[x] > _par[y]) {
            swap(x, y);
            a = 1 / a;
            b *= -a;
        }
        if(_data[y].vertex_value_determined) {
            if(_data[x].vertex_value_determined) {
                if(a * _data[x].vertex_value + b != _data[y].vertex_value) return {false, false};
            } else {
                _data[x].vertex_value_determined = true;
                _data[x].vertex_value = (_data[y].vertex_value - b) / a;
            }
        }
        _par[x] += _par[y];
        _par[y] = x;
        if constexpr (!is_same_v<GroupValue, nullptr_t>) {
            _data[x].group_value = _merge_group_value(_data[x].group_value, _data[y].group_value);
        }
        _data[y].a = a;
        _data[y].b = b;
        return {true, true};
    }
    bool same(int x, int y) {
        return root(x) == root(y);
    }
    VertexValue diff(int x, int y) {
        static_assert(!is_same_v<VertexValue, nullptr_t>);
        static_assert(is_same_v<VertexValueCoef, nullptr_t>);
        assert(root(x) == root(y));
        return _data[y].b - _data[x].b;
    }
    pair<VertexValueCoef, VertexValue> relation(int x, int y) {
        static_assert(!is_same_v<VertexValue, nullptr_t>);
        static_assert(!is_same_v<VertexValueCoef, nullptr_t>);
        assert(root(x) == root(y));
        VertexValueCoef a = _data[y].a / _data[x].a;
        VertexValue b = _data[y].b - a * _data[x].b;
        return {a, b};
    }
    VVI groups() {
        VVI res(_n);
        REP(i, _n) {
            res[root(i)].push_back(i);
        }
        sort(ALL(res), [](const VI &a, const VI &b) { return a.size() > b.size(); });
        while (res.size() && res.back().empty()) res.pop_back();
        return res;
    }
};

template <typename Potential>
using PotentialUnionFind = UnionFind<nullptr_t, nullptr_t, Potential>;
template <typename Coef, typename Potential>
using RelationUnionFind = UnionFind<nullptr_t, nullptr_t, Potential, Coef>;
#line 3 "Contests/yukicoder_381/yukicoder_381_f/main.cpp"

/* #include <atcoder/all> */
/* using namespace atcoder; */
/* #include "KowerKoint/expansion/ac-library/all.hpp" */

void solve(){
    int h, w; cin >> h >> w;
    VVI a(h, VI(w)); cin >> a;
    map<int, VP> mp;
    REP(i, h) REP(j, w) {
        mp[-a[i][j]].push_back(P(i, j));
    }
    V3 two_pow(h+w+1);
    two_pow[0] = 1;
    REP(i, h+w) two_pow[i+1] = two_pow[i] * 2;
    int groups = h+w;
    UnionFind uf(h+w);
    MI3 ans = 0;
    vector<pair<int, MI3>> hoge;
    for(auto& [k, v] : mp) {
        hoge.emplace_back(-k, two_pow[groups-1]);
        for(auto [x, y] : v) {
            if(uf.merge(x, y+h)) groups--;
        }
    }
    hoge.emplace_back(0, two_pow[groups-1]);
    REP(i, (int)hoge.size()-1) ans += MI3(hoge[i].first) * (hoge[i].second - hoge[i+1].second);
    print(ans);
}

// generated by oj-template v4.7.2 (https://github.com/online-judge-tools/template-generator)
int main() {
    // Fasterize input/output script
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout << fixed << setprecision(100);
    // scanf/printf user should delete this fasterize input/output script

    int t = 1;
    //cin >> t; // comment out if solving multi testcase
    for(int testCase = 1;testCase <= t;++testCase){
        solve();
    }
    return 0;
}