#include <bits/stdc++.h>
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using f64 = double;
using f80 = long double;
using f128 = __float128;
constexpr i32 operator"" _i32(u64 v) { return v; }
constexpr u32 operator"" _u32(u64 v) { return v; }
constexpr i64 operator"" _i64(u64 v) { return v; }
constexpr u64 operator"" _u64(u64 v) { return v; }
constexpr f64 operator"" _f64(f80 v) { return v; }
constexpr f80 operator"" _f80(f80 v) { return v; }
using Istream = std::istream;
using Ostream = std::ostream;
using Str = std::string;
template<typename T>
using Lt = std::less<T>;
template<typename T>
using Gt = std::greater<T>;
template<int n>
using BSet = std::bitset<n>;
template<typename T1, typename T2>
using Pair = std::pair<T1, T2>;
template<typename... Ts>
using Tup = std::tuple<Ts...>;
template<typename T, int N>
using Arr = std::array<T, N>;
template<typename... Ts>
using Deq = std::deque<Ts...>;
template<typename... Ts>
using Set = std::set<Ts...>;
template<typename... Ts>
using MSet = std::multiset<Ts...>;
template<typename... Ts>
using USet = std::unordered_set<Ts...>;
template<typename... Ts>
using UMSet = std::unordered_multiset<Ts...>;
template<typename... Ts>
using Map = std::map<Ts...>;
template<typename... Ts>
using MMap = std::multimap<Ts...>;
template<typename... Ts>
using UMap = std::unordered_map<Ts...>;
template<typename... Ts>
using UMMap = std::unordered_multimap<Ts...>;
template<typename... Ts>
using Vec = std::vector<Ts...>;
template<typename... Ts>
using Stack = std::stack<Ts...>;
template<typename... Ts>
using Queue = std::queue<Ts...>;
template<typename T>
using MaxHeap = std::priority_queue<T>;
template<typename T>
using MinHeap = std::priority_queue<T, Vec<T>, Gt<T>>;
constexpr bool LOCAL = false;
constexpr bool OJ = not LOCAL;
template<typename T>
static constexpr T OjLocal(T oj, T local)
{
    return LOCAL ? local : oj;
}
template<typename T>
constexpr T LIMMIN = std::numeric_limits<T>::min();
template<typename T>
constexpr T LIMMAX = std::numeric_limits<T>::max();
template<typename T>
constexpr T INF = (LIMMAX<T> - 1) / 2;
template<typename T>
constexpr T PI = T{3.141592653589793238462643383279502884};
template<typename T = u64>
constexpr T TEN(int n)
{
    return n == 0 ? T{1} : TEN<T>(n - 1) * T{10};
}
template<typename T>
constexpr bool chmin(T& a, const T& b)
{
    return (a > b ? (a = b, true) : false);
}
template<typename T>
constexpr bool chmax(T& a, const T& b)
{
    return (a < b ? (a = b, true) : false);
}
template<typename T>
constexpr T floorDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? x / y : (x - y + 1) / y;
}
template<typename T>
constexpr T ceilDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? (x + y - 1) / y : x / y;
}
template<typename T, typename I>
constexpr T powerMonoid(T v, I n, const T& e)
{
    assert(n >= 0);
    if (n == 0) { return e; }
    return (n % 2 == 1 ? v * powerMonoid(v, n - 1, e) : powerMonoid(v * v, n / 2, e));
}
template<typename T, typename I>
constexpr T powerInt(T v, I n)
{
    return powerMonoid(v, n, T{1});
}
template<typename Vs, typename V>
constexpr void fillAll(Vs& arr, const V& v)
{
    if constexpr (std::is_convertible<V, Vs>::value) {
        arr = v;
    } else {
        for (auto& subarr : arr) { fillAll(subarr, v); }
    }
}
template<typename Vs>
constexpr void sortAll(Vs& vs)
{
    std::sort(std::begin(vs), std::end(vs));
}
template<typename Vs, typename C>
constexpr void sortAll(Vs& vs, C comp)
{
    std::sort(std::begin(vs), std::end(vs), comp);
}
template<typename Vs>
constexpr void reverseAll(Vs& vs)
{
    std::reverse(std::begin(vs), std::end(vs));
}
template<typename Vs>
constexpr Vs reversed(const Vs& vs)
{
    auto rvs = vs;
    reverseAll(rvs);
    return rvs;
}
template<typename V, typename Vs>
constexpr V sumAll(const Vs& vs)
{
    if constexpr (std::is_convertible<Vs, V>::value) {
        return static_cast<V>(vs);
    } else {
        V ans = 0;
        for (const auto& v : vs) { ans += sumAll<V>(v); }
        return ans;
    }
}
template<typename Vs>
constexpr int minInd(const Vs& vs)
{
    return std::min_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs>
constexpr int maxInd(const Vs& vs)
{
    return std::max_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr int lbInd(const Vs& vs, const V& v)
{
    return std::lower_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr int ubInd(const Vs& vs, const V& v)
{
    return std::upper_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr void plusAll(Vs& vs, const V& v)
{
    for (auto& v_ : vs) { v_ += v; }
}
template<typename Vs>
constexpr void concat(Vs& vs1, const Vs& vs2)
{
    std::copy(std::begin(vs2), std::end(vs2), std::back_inserter(vs1));
}
template<typename Vs>
constexpr void concatted(const Vs& vs1, const Vs& vs2)
{
    auto vs = vs1;
    concat(vs, vs2);
    return vs;
}
template<typename T, typename F>
constexpr Vec<T> genVec(int n, F gen)
{
    Vec<T> ans;
    std::generate_n(std::back_inserter(ans), n, gen);
    return ans;
}
template<typename T = int>
constexpr Vec<T> iotaVec(int n, T offset = 0)
{
    Vec<T> ans(n);
    std::iota(std::begin(ans), std::end(ans), offset);
    return ans;
}
template<typename Vs>
constexpr void rearrange(Vs& vs, const Vec<int>& is)
{
    auto vs_ = vs;
    for (int i = 0; i < (int)is.size(); i++) { vs[i] = vs_[is[i]]; }
}
inline Vec<int> reversePerm(const Vec<int>& is)
{
    auto ris = is;
    for (int i = 0; i < (int)is.size(); i++) { ris[is[i]] = i; }
    return ris;
}
inline Ostream& operator<<(Ostream& os, i128 v)
{
    bool minus = false;
    if (v < 0) { minus = true, v = -v; }
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << (minus ? "-" : "") << ans;
}
inline Ostream& operator<<(Ostream& os, u128 v)
{
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << ans;
}
constexpr int popCount(u64 v) { return v ? __builtin_popcountll(v) : 0; }
constexpr int topBit(u64 v) { return v == 0 ? -1 : 63 - __builtin_clzll(v); }
constexpr int lowBit(u64 v) { return v == 0 ? 64 : __builtin_ctzll(v); }
constexpr int bitWidth(u64 v) { return topBit(v) + 1; }
constexpr u64 bitCeil(u64 v) { return v ? (1_u64 << bitWidth(v - 1)) : 1_u64; }
constexpr u64 bitFloor(u64 v) { return v ? (1_u64 << topBit(v)) : 0_u64; }
constexpr bool hasSingleBit(u64 v) { return (v > 0) and ((v & (v - 1)) == 0); }
constexpr bool isBitOn(u64 mask, int ind) { return (mask >> ind) & 1_u64; }
constexpr bool isBitOff(u64 mask, int ind) { return not isBitOn(mask, ind); }
constexpr u64 bitMask(int bitWidth) { return (bitWidth == 64 ? ~0_u64 : (1_u64 << bitWidth) - 1); }
constexpr u64 bitMask(int start, int end) { return bitMask(end - start) << start; }
template<typename F>
struct Fix : F
{
    constexpr Fix(F&& f) : F{std::forward<F>(f)} {}
    template<typename... Args>
    constexpr auto operator()(Args&&... args) const
    {
        return F::operator()(*this, std::forward<Args>(args)...);
    }
};
class irange
{
private:
    struct itr
    {
        constexpr itr(i64 start = 0, i64 step = 1) : m_cnt{start}, m_step{step} {}
        constexpr bool operator!=(const itr& it) const { return m_cnt != it.m_cnt; }
        constexpr i64 operator*() { return m_cnt; }
        constexpr itr& operator++() { return m_cnt += m_step, *this; }
        i64 m_cnt, m_step;
    };
    i64 m_start, m_end, m_step;
public:
    static constexpr i64 cnt(i64 start, i64 end, i64 step)
    {
        if (step == 0) { return -1; }
        const i64 d = (step > 0 ? step : -step);
        const i64 l = (step > 0 ? start : end);
        const i64 r = (step > 0 ? end : start);
        i64 n = (r - l) / d + ((r - l) % d ? 1 : 0);
        if (l >= r) { n = 0; }
        return n;
    }
    constexpr irange(i64 start, i64 end, i64 step = 1)
        : m_start{start}, m_end{m_start + step * cnt(start, end, step)}, m_step{step}
    {
        assert(step != 0);
    }
    constexpr itr begin() const { return itr{m_start, m_step}; }
    constexpr itr end() const { return itr{m_end, m_step}; }
};
constexpr irange rep(i64 end) { return irange(0, end, 1); }
constexpr irange per(i64 rend) { return irange(rend - 1, -1, -1); }
class Scanner
{
public:
    Scanner(Istream& is = std::cin) : m_is{is} { m_is.tie(nullptr)->sync_with_stdio(false); }
    template<typename T>
    T val()
    {
        T v;
        return m_is >> v, v;
    }
    template<typename T>
    T val(T offset)
    {
        return val<T>() - offset;
    }
    template<typename T>
    Vec<T> vec(int n)
    {
        return genVec<T>(n, [&]() { return val<T>(); });
    }
    template<typename T>
    Vec<T> vec(int n, T offset)
    {
        return genVec<T>(n, [&]() { return val<T>(offset); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, const T offset)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m, offset); });
    }
    template<typename... Args>
    auto tup()
    {
        return Tup<Args...>{val<Args>()...};
    }
    template<typename... Args>
    auto tup(const Args&... offsets)
    {
        return Tup<Args...>{val<Args>(offsets)...};
    }
private:
    Istream& m_is;
};
inline Scanner in;
class Printer
{
public:
    Printer(Ostream& os = std::cout) : m_os{os} { m_os << std::fixed << std::setprecision(15); }
    template<typename... Args>
    int operator()(const Args&... args)
    {
        return dump(args...), 0;
    }
    template<typename... Args>
    int ln(const Args&... args)
    {
        return dump(args...), m_os << '\n', 0;
    }
    template<typename... Args>
    int el(const Args&... args)
    {
        return dump(args...), m_os << std::endl, 0;
    }
    int YES(bool b = true) { return ln(b ? "YES" : "NO"); }
    int NO(bool b = true) { return YES(not b); }
    int Yes(bool b = true) { return ln(b ? "Yes" : "No"); }
    int No(bool b = true) { return Yes(not b); }
private:
    template<typename T>
    void dump(const T& v)
    {
        m_os << v;
    }
    template<typename T>
    void dump(const Vec<T>& vs)
    {
        for (int i : rep(vs.size())) { m_os << (i ? " " : ""), dump(vs[i]); }
    }
    template<typename T>
    void dump(const Vec<Vec<T>>& vss)
    {
        for (int i : rep(vss.size())) { m_os << (i ? "\n" : ""), dump(vss[i]); }
    }
    template<typename T, typename... Ts>
    int dump(const T& v, const Ts&... args)
    {
        return dump(v), m_os << ' ', dump(args...), 0;
    }
    Ostream& m_os;
};
inline Printer out;
template<typename T, int n, int i = 0>
auto ndVec(int const (&szs)[n], const T x = T{})
{
    if constexpr (i == n) {
        return x;
    } else {
        return std::vector(szs[i], ndVec<T, n, i + 1>(szs, x));
    }
}
template<typename T, typename F>
inline T binSearch(T ng, T ok, F check)
{
    while (std::abs(ok - ng) > 1) {
        const T mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template<typename T>
constexpr Pair<T, T> extgcd(const T a, const T b) // [x,y] -> ax+by=gcd(a,b)
{
    static_assert(std::is_signed_v<T>, "Signed integer is allowed.");
    assert(a != 0 or b != 0);
    if (a >= 0 and b >= 0) {
        if (a < b) {
            const auto [y, x] = extgcd(b, a);
            return {x, y};
        }
        if (b == 0) { return {1, 0}; }
        const auto [x, y] = extgcd(b, a % b);
        return {y, x - (a / b) * y};
    } else {
        auto [x, y] = extgcd(std::abs(a), std::abs(b));
        if (a < 0) { x = -x; }
        if (b < 0) { y = -y; }
        return {x, y};
    }
}
template<typename T>
constexpr T inverse(const T a, const T mod) // ax=gcd(a,M) (mod M)
{
    assert(a > 0 and mod > 0);
    auto [x, y] = extgcd(a, mod);
    if (x <= 0) { x += mod; }
    return x;
}
template<u32 mod_, u32 root_, u32 max2p_>
class modint
{
    template<typename U = u32&>
    static U modRef()
    {
        static u32 s_mod = 0;
        return s_mod;
    }
    template<typename U = u32&>
    static U rootRef()
    {
        static u32 s_root = 0;
        return s_root;
    }
    template<typename U = u32&>
    static U max2pRef()
    {
        static u32 s_max2p = 0;
        return s_max2p;
    }
public:
    static_assert(mod_ <= LIMMAX<i32>, "mod(signed int size) only supported!");
    static constexpr bool isDynamic() { return (mod_ == 0); }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u32>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        assert(1 <= m and m <= LIMMAX<i32>);
        modRef() = m;
        sinvRef() = {1, 1};
        factRef() = {1, 1};
        ifactRef() = {1, 1};
    }
    template<typename U = u32>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u32>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint() : m_val{0} {}
    constexpr modint(i64 v) : m_val{normll(v)} {}
    constexpr void setRaw(u32 v) { m_val = v; }
    constexpr modint operator-() const { return modint{0} - (*this); }
    constexpr modint& operator+=(const modint& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint& operator-=(const modint& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint& operator*=(const modint& m)
    {
        m_val = normll((i64)m_val * (i64)m.val() % (i64)mod());
        return *this;
    }
    constexpr modint& operator/=(const modint& m) { return *this *= m.inv(); }
    constexpr modint operator+(const modint& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint operator-(const modint& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint operator*(const modint& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint operator/(const modint& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint& m) const { return m_val == m.val(); }
    constexpr bool operator!=(const modint& m) const { return not(*this == m); }
    friend Istream& operator>>(Istream& is, modint& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint& m) { return os << m.val(); }
    constexpr u32 val() const { return m_val; }
    template<typename I>
    constexpr modint pow(I n) const
    {
        return powerInt(*this, n);
    }
    constexpr modint inv() const { return inverse<i32>(m_val, mod()); }
    static modint sinv(u32 n)
    {
        auto& is = sinvRef();
        for (u32 i = (u32)is.size(); i <= n; i++) { is.push_back(-is[mod() % i] * (mod() / i)); }
        return is[n];
    }
    static modint fact(u32 n)
    {
        auto& fs = factRef();
        for (u32 i = (u32)fs.size(); i <= n; i++) { fs.push_back(fs.back() * i); }
        return fs[n];
    }
    static modint ifact(u32 n)
    {
        auto& ifs = ifactRef();
        for (u32 i = (u32)ifs.size(); i <= n; i++) { ifs.push_back(ifs.back() * sinv(i)); }
        return ifs[n];
    }
    static modint perm(int n, int k) { return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k); }
    static modint comb(int n, int k)
    {
        return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static Vec<modint>& sinvRef()
    {
        static Vec<modint> is{1, 1};
        return is;
    }
    static Vec<modint>& factRef()
    {
        static Vec<modint> fs{1, 1};
        return fs;
    }
    static Vec<modint>& ifactRef()
    {
        static Vec<modint> ifs{1, 1};
        return ifs;
    }
    static constexpr u32 norm(u32 x) { return x < mod() ? x : x - mod(); }
    static constexpr u32 normll(i64 x) { return norm(u32(x % (i64)mod() + (i64)mod())); }
    u32 m_val;
};
using modint_1000000007 = modint<1000000007, 5, 1>;
using modint_998244353 = modint<998244353, 3, 23>;
template<int id>
using modint_dynamic = modint<0, 0, id>;
template<typename T = int>
class Graph
{
    struct Edge
    {
        Edge() = default;
        Edge(int i, int t, T c) : id{i}, to{t}, cost{c} {}
        int id;
        int to;
        T cost;
        operator int() const { return to; }
    };
public:
    Graph(int n) : m_v{n}, m_edges(n) {}
    void addEdge(int u, int v, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, 1);
        if (bi) { m_edges[v].emplace_back(m_e, u, 1); }
        m_e++;
    }
    void addEdge(int u, int v, const T& c, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, c);
        if (bi) { m_edges[v].emplace_back(m_e, u, c); }
        m_e++;
    }
    const Vec<Edge>& operator[](const int u) const
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    Vec<Edge>& operator[](const int u)
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    int v() const { return m_v; }
    int e() const { return m_e; }
    friend Ostream& operator<<(Ostream& os, const Graph& g)
    {
        for (int u : rep(g.v())) {
            for (const auto& [id, v, c] : g[u]) {
                os << "[" << id << "]: ";
                os << u << "->" << v << "(" << c << ")\n";
            }
        }
        return os;
    }
    Vec<T> sizes(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ss(N, 1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_0, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                dfs(v, u);
                ss[u] += ss[v];
            }
        })(root, -1);
        return ss;
    }
    Vec<T> depths(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ds(N, 0);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_1, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ds[v] = ds[u] + c;
                dfs(v, u);
            }
        })(root, -1);
        return ds;
    }
    Vec<int> parents(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<int> ps(N, -1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_2, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ps[v] = u;
                dfs(v, u);
            }
        })(root, -1);
        return ps;
    }
private:
    int m_v;
    int m_e = 0;
    Vec<Vec<Edge>> m_edges;
};
using namespace std;
struct UnionFind
{
    vector<int> data;
    UnionFind() = default;
    explicit UnionFind(size_t sz) : data(sz, -1) {}
    bool unite(int x, int y)
    {
        x = find(x), y = find(y);
        if (x == y)
            return false;
        if (data[x] > data[y])
            swap(x, y);
        data[x] += data[y];
        data[y] = x;
        return true;
    }
    int find(int k)
    {
        if (data[k] < 0)
            return (k);
        return data[k] = find(data[k]);
    }
    int size(int k)
    {
        return -data[find(k)];
    }
    bool same(int x, int y)
    {
        return find(x) == find(y);
    }
    vector<vector<int>> groups()
    {
        int n = (int)data.size();
        vector<vector<int>> ret(n);
        for (int i = 0; i < n; i++) {
            ret[find(i)].emplace_back(i);
        }
        ret.erase(remove_if(begin(ret),
                            end(ret),
                            [&](const vector<int>& v) { return v.empty(); }),
                  end(ret));
        return ret;
    }
};
/**
 * @brief Bipartite Flow(二部グラフのフロー)
 * @docs docs/bipartite-flow.md
 */
struct BipartiteFlow
{
    size_t n, m, time_stamp;
    vector<vector<int>> g, rg;
    vector<int> match_l, match_r, dist, used, alive;
    bool matched;
public:
    explicit BipartiteFlow(size_t n, size_t m)
        : n(n),
          m(m),
          time_stamp(0),
          g(n),
          rg(m),
          match_l(n, -1),
          match_r(m, -1),
          used(n),
          alive(n, 1),
          matched(false)
    {}
    void add_edge(int u, int v)
    {
        g[u].push_back(v);
        rg[v].emplace_back(u);
    }
    vector<pair<int, int>> max_matching()
    {
        matched = true;
        for (;;) {
            build_augment_path();
            ++time_stamp;
            int flow = 0;
            for (int i = 0; i < (int)n; i++) {
                if (match_l[i] == -1)
                    flow += find_min_dist_augment_path(i);
            }
            if (flow == 0)
                break;
        }
        vector<pair<int, int>> ret;
        for (int i = 0; i < (int)n; i++) {
            if (match_l[i] >= 0)
                ret.emplace_back(i, match_l[i]);
        }
        return ret;
    }
    /* http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=3198 */
    void erase_edge(int a, int b)
    {
        if (match_l[a] == b) {
            match_l[a] = -1;
            match_r[b] = -1;
        }
        g[a].erase(find(begin(g[a]), end(g[a]), b));
        rg[b].erase(find(begin(rg[b]), end(rg[b]), a));
    }
    /* http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=0334 */
    vector<pair<int, int>> lex_max_matching()
    {
        if (!matched)
            max_matching();
        for (auto& vs : g)
            sort(begin(vs), end(vs));
        vector<pair<int, int>> es;
        for (int i = 0; i < (int)n; i++) {
            if (match_l[i] == -1 || alive[i] == 0) {
                continue;
            }
            match_r[match_l[i]] = -1;
            match_l[i] = -1;
            ++time_stamp;
            find_augment_path(i);
            alive[i] = 0;
            es.emplace_back(i, match_l[i]);
        }
        return es;
    }
    vector<int> min_vertex_cover()
    {
        auto visited = find_residual_path();
        vector<int> ret;
        for (int i = 0; i < (int)(n + m); i++) {
            if (visited[i] ^ (i < (int)n)) {
                ret.emplace_back(i);
            }
        }
        return ret;
    }
    /* https://atcoder.jp/contests/utpc2013/tasks/utpc2013_11 */
    vector<int> lex_min_vertex_cover(const vector<int>& ord)
    {
        assert(ord.size() == n + m);
        auto res = build_risidual_graph();
        vector<vector<int>> r_res(n + m + 2);
        for (int i = 0; i < (int)(n + m + 2); i++) {
            for (auto& j : res[i])
                r_res[j].emplace_back(i);
        }
        queue<int> que;
        vector<int> visited(n + m + 2, -1);
        auto expand_left = [&](int t) {
            if (visited[t] != -1)
                return;
            que.emplace(t);
            visited[t] = 1;
            while (!que.empty()) {
                int idx = que.front();
                que.pop();
                for (auto& to : r_res[idx]) {
                    if (visited[to] != -1)
                        continue;
                    visited[to] = 1;
                    que.emplace(to);
                }
            }
        };
        auto expand_right = [&](int t) {
            if (visited[t] != -1)
                return;
            que.emplace(t);
            visited[t] = 0;
            while (!que.empty()) {
                int idx = que.front();
                que.pop();
                for (auto& to : res[idx]) {
                    if (visited[to] != -1)
                        continue;
                    visited[to] = 0;
                    que.emplace(to);
                }
            }
        };
        expand_right(n + m);
        expand_left(n + m + 1);
        vector<int> ret;
        for (auto& t : ord) {
            if (t < (int)n) {
                expand_left(t);
                if (visited[t] & 1)
                    ret.emplace_back(t);
            } else {
                expand_right(t);
                if (~visited[t] & 1)
                    ret.emplace_back(t);
            }
        }
        return ret;
    }
    vector<int> max_independent_set()
    {
        auto visited = find_residual_path();
        vector<int> ret;
        for (int i = 0; i < (int)(n + m); i++) {
            if (visited[i] ^ (i >= (int)n)) {
                ret.emplace_back(i);
            }
        }
        return ret;
    }
    vector<pair<int, int>> min_edge_cover()
    {
        auto es = max_matching();
        for (int i = 0; i < (int)n; i++) {
            if (match_l[i] >= 0) {
                continue;
            }
            if (g[i].empty()) {
                return {};
            }
            es.emplace_back(i, g[i][0]);
        }
        for (int i = 0; i < (int)m; i++) {
            if (match_r[i] >= 0) {
                continue;
            }
            if (rg[i].empty()) {
                return {};
            }
            es.emplace_back(rg[i][0], i);
        }
        return es;
    }
    // left: [0,n), right: [n,n+m), S: n+m, T: n+m+1
    vector<vector<int>> build_risidual_graph()
    {
        if (!matched)
            max_matching();
        const size_t S = n + m;
        const size_t T = n + m + 1;
        vector<vector<int>> ris(n + m + 2);
        for (int i = 0; i < (int)n; i++) {
            if (match_l[i] == -1)
                ris[S].emplace_back(i);
            else
                ris[i].emplace_back(S);
        }
        for (int i = 0; i < (int)m; i++) {
            if (match_r[i] == -1)
                ris[i + n].emplace_back(T);
            else
                ris[T].emplace_back(i + n);
        }
        for (int i = 0; i < (int)n; i++) {
            for (auto& j : g[i]) {
                if (match_l[i] == j)
                    ris[j + n].emplace_back(i);
                else
                    ris[i].emplace_back(j + n);
            }
        }
        return ris;
    }
private:
    vector<int> find_residual_path()
    {
        auto res = build_risidual_graph();
        queue<int> que;
        vector<int> visited(n + m + 2);
        que.emplace(n + m);
        visited[n + m] = true;
        while (!que.empty()) {
            int idx = que.front();
            que.pop();
            for (auto& to : res[idx]) {
                if (visited[to])
                    continue;
                visited[to] = true;
                que.emplace(to);
            }
        }
        return visited;
    }
    void build_augment_path()
    {
        queue<int> que;
        dist.assign(g.size(), -1);
        for (int i = 0; i < (int)n; i++) {
            if (match_l[i] == -1) {
                que.emplace(i);
                dist[i] = 0;
            }
        }
        while (!que.empty()) {
            int a = que.front();
            que.pop();
            for (auto& b : g[a]) {
                int c = match_r[b];
                if (c >= 0 && dist[c] == -1) {
                    dist[c] = dist[a] + 1;
                    que.emplace(c);
                }
            }
        }
    }
    bool find_min_dist_augment_path(int a)
    {
        used[a] = time_stamp;
        for (auto& b : g[a]) {
            int c = match_r[b];
            if (c < 0
                || (used[c] != (int)time_stamp && dist[c] == dist[a] + 1
                    && find_min_dist_augment_path(c))) {
                match_r[b] = a;
                match_l[a] = b;
                return true;
            }
        }
        return false;
    }
    bool find_augment_path(int a)
    {
        used[a] = time_stamp;
        for (auto& b : g[a]) {
            int c = match_r[b];
            if (c < 0
                || (alive[c] == 1 && used[c] != (int)time_stamp
                    && find_augment_path(c))) {
                match_r[b] = a;
                match_l[a] = b;
                return true;
            }
        }
        return false;
    }
};
/**
 * @brief Eulerian Trail(オイラー路)
 * @docs docs/eulerian-trail.md
 */
template<bool directed>
struct EulerianTrail
{
    vector<vector<pair<int, int>>> g;
    vector<pair<int, int>> es;
    int M;
    vector<int> used_vertex, used_edge, deg;
    explicit EulerianTrail(int V) : g(V), M(0), used_vertex(V), deg(V) {}
    void add_edge(int a, int b)
    {
        es.emplace_back(a, b);
        g[a].emplace_back(b, M);
        if (directed) {
            deg[a]++;
            deg[b]--;
        } else {
            g[b].emplace_back(a, M);
            deg[a]++;
            deg[b]++;
        }
        M++;
    }
    pair<int, int> get_edge(int idx) const
    {
        return es[idx];
    }
    vector<vector<int>> enumerate_eulerian_trail()
    {
        if (directed) {
            for (auto& p : deg)
                if (p != 0)
                    return {};
        } else {
            for (auto& p : deg)
                if (p & 1)
                    return {};
        }
        used_edge.assign(M, 0);
        vector<vector<int>> ret;
        for (int i = 0; i < (int)g.size(); i++) {
            if (g[i].empty() || used_vertex[i])
                continue;
            ret.emplace_back(go(i));
        }
        return ret;
    }
    vector<vector<int>> enumerate_semi_eulerian_trail()
    {
        UnionFind uf(g.size());
        for (auto& p : es)
            uf.unite(p.first, p.second);
        vector<vector<int>> group(g.size());
        for (int i = 0; i < (int)g.size(); i++)
            group[uf.find(i)].emplace_back(i);
        vector<vector<int>> ret;
        used_edge.assign(M, 0);
        for (auto& vs : group) {
            if (vs.empty())
                continue;
            int latte = -1, malta = -1;
            if (directed) {
                for (auto& p : vs) {
                    if (abs(deg[p]) > 1) {
                        return {};
                    } else if (deg[p] == 1) {
                        if (latte >= 0)
                            return {};
                        latte = p;
                    }
                }
            } else {
                for (auto& p : vs) {
                    if (deg[p] & 1) {
                        if (latte == -1)
                            latte = p;
                        else if (malta == -1)
                            malta = p;
                        else
                            return {};
                    }
                }
            }
            ret.emplace_back(go(latte == -1 ? vs.front() : latte));
            if (ret.back().empty())
                ret.pop_back();
        }
        return ret;
    }
    vector<int> go(int s)
    {
        stack<pair<int, int>> st;
        vector<int> ord;
        st.emplace(s, -1);
        while (!st.empty()) {
            int idx = st.top().first;
            used_vertex[idx] = true;
            if (g[idx].empty()) {
                ord.emplace_back(st.top().second);
                st.pop();
            } else {
                auto e = g[idx].back();
                g[idx].pop_back();
                if (used_edge[e.second])
                    continue;
                used_edge[e.second] = true;
                st.emplace(e);
            }
        }
        ord.pop_back();
        reverse(ord.begin(), ord.end());
        return ord;
    }
};
/**
 * @brief Bipartite Graph Edge Coloring(二部グラフの辺彩色)
 * @docs docs/bipartite-graph-edge-coloring.md
 * @see https://ei1333.hateblo.jp/entry/2020/08/25/015955
 */
struct BipariteGraphEdgeColoring
{
private:
    vector<vector<int>> ans;
    vector<int> A, B;
    int L, R;
    struct RegularGraph
    {
        int k{}, n{};
        vector<int> A, B;
    };
    RegularGraph g;
    static UnionFind contract(valarray<int>& deg, int k)
    {
        using pi = pair<int, int>;
        priority_queue<pi, vector<pi>, greater<>> que;
        for (int i = 0; i < (int)deg.size(); i++) {
            que.emplace(deg[i], i);
        }
        UnionFind uf(deg.size());
        while (que.size() > 1) {
            auto p = que.top();
            que.pop();
            auto q = que.top();
            que.pop();
            if (p.first + q.first > k)
                continue;
            p.first += q.first;
            uf.unite(p.second, q.second);
            que.emplace(p);
        }
        return uf;
    }
    RegularGraph build_k_regular_graph()
    {
        valarray<int> deg[2];
        deg[0] = valarray<int>(L);
        deg[1] = valarray<int>(R);
        for (auto& p : A)
            deg[0][p]++;
        for (auto& p : B)
            deg[1][p]++;
        int k = max(deg[0].max(), deg[1].max());
        /* step 1 */
        UnionFind uf[2];
        uf[0] = contract(deg[0], k);
        uf[1] = contract(deg[1], k);
        vector<int> id[2];
        int ptr[] = {0, 0};
        id[0] = vector<int>(L);
        id[1] = vector<int>(R);
        for (int i = 0; i < L; i++)
            if (uf[0].find(i) == i)
                id[0][i] = ptr[0]++;
        for (int i = 0; i < R; i++)
            if (uf[1].find(i) == i)
                id[1][i] = ptr[1]++;
        /* step 2 */
        int N = max(ptr[0], ptr[1]);
        deg[0] = valarray<int>(N);
        deg[1] = valarray<int>(N);
        /* step 3 */
        vector<int> C, D;
        C.reserve(N * k);
        D.reserve(N * k);
        for (int i = 0; i < (int)A.size(); i++) {
            int u = id[0][uf[0].find(A[i])];
            int v = id[1][uf[1].find(B[i])];
            C.emplace_back(u);
            D.emplace_back(v);
            deg[0][u]++;
            deg[1][v]++;
        }
        int j = 0;
        for (int i = 0; i < N; i++) {
            while (deg[0][i] < k) {
                while (deg[1][j] == k)
                    ++j;
                C.emplace_back(i);
                D.emplace_back(j);
                ++deg[0][i];
                ++deg[1][j];
            }
        }
        return {k, N, C, D};
    }
    void rec(const vector<int>& ord, int k)
    {
        if (k == 0) {
            return;
        } else if (k == 1) {
            ans.emplace_back(ord);
            return;
        } else if ((k & 1) == 0) {
            EulerianTrail<false> et(g.n + g.n);
            for (auto& p : ord)
                et.add_edge(g.A[p], g.B[p] + g.n);
            auto paths = et.enumerate_eulerian_trail();
            vector<int> path;
            for (auto& ps : paths) {
                for (auto& e : ps)
                    path.emplace_back(ord[e]);
            }
            vector<int> beet[2];
            for (int i = 0; i < (int)path.size(); i++) {
                beet[i & 1].emplace_back(path[i]);
            }
            rec(beet[0], k / 2);
            rec(beet[1], k / 2);
        } else {
            BipartiteFlow flow(g.n, g.n);
            for (auto& i : ord)
                flow.add_edge(g.A[i], g.B[i]);
            flow.max_matching();
            vector<int> beet;
            ans.emplace_back();
            for (auto& i : ord) {
                if (flow.match_l[g.A[i]] == g.B[i]) {
                    flow.match_l[g.A[i]] = -1;
                    ans.back().emplace_back(i);
                } else {
                    beet.emplace_back(i);
                }
            }
            rec(beet, k - 1);
        }
    }
public:
    explicit BipariteGraphEdgeColoring() : L(0), R(0) {}
    void add_edge(int a, int b)
    {
        A.emplace_back(a);
        B.emplace_back(b);
        L = max(L, a + 1);
        R = max(R, b + 1);
    }
    vector<vector<int>> build()
    {
        g = build_k_regular_graph();
        vector<int> ord(g.A.size());
        iota(ord.begin(), ord.end(), 0);
        rec(ord, g.k);
        vector<vector<int>> res;
        for (int i = 0; i < (int)ans.size(); i++) {
            res.emplace_back();
            for (auto& j : ans[i])
                if (j < (int)A.size())
                    res.back().emplace_back(j);
        }
        return res;
    }
};
int main()
{
    const auto [N, M] = in.tup<int, int>();
    const auto Ass = in.vvec<int>(N, N);
    Vec<int> lds(N,0), rds(N, 0);
    Vec<Pair<int, int>> es;
    for (int i : rep(N)) {
        for (int j : rep(N)) {
            lds[i] += Ass[i][j];
            rds[j] += Ass[i][j];
        }
    }
    for (int i : rep(N)) {
        if (lds[i] != M) {
            return out.ln(-1);
        }
        if (rds[i] != M) {
            return out.ln(-1);
        }
    }
    BipariteGraphEdgeColoring g;
    for (int i : rep(N)) {
        for (int j : rep(N)) {
            for (auto _temp_name_3 [[maybe_unused]] : rep(Ass[i][j])) {
                g.add_edge(i, j);
                es.push_back({i, j});
            }
        }
    }
    const auto vss = g.build();
    for (const auto& vs : vss) {
        Vec<int> Ps(N, -1);
        for (int ei : vs) {
            const auto [i, j] = es[ei];
            Ps[i] = j;
        }
        plusAll(Ps, 1);
        out.ln(Ps);
    }
    return 0;
}