#line 1 "ucup3-16/F/main.cpp"
#define YOSUPO_AVX2_PRAGMA
// #undef YOSUPO_LOCAL

#line 2 "/home/vscode/yosupo-library/src/yosupo/fastio.hpp"

#include <unistd.h>
#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <cctype>
#include <cstdint>
#include <cstring>
#include <sstream>
#include <string>
#include <type_traits>
#include <vector>

#line 2 "/home/vscode/yosupo-library/src/yosupo/internal_type_traits.hpp"

#line 5 "/home/vscode/yosupo-library/src/yosupo/internal_type_traits.hpp"

namespace yosupo {

namespace internal {

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

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

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

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

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class 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;

template <class T>
using is_integral_t = std::enable_if_t<is_integral<T>::value>;

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace yosupo
#line 17 "/home/vscode/yosupo-library/src/yosupo/fastio.hpp"

namespace yosupo {

struct Scanner {
  public:
    Scanner(const Scanner&) = delete;
    Scanner& operator=(const Scanner&) = delete;

    Scanner(FILE* fp) : fd(fileno(fp)) { line[0] = 127; }

    void read() {}
    template <class H, class... T> void read(H& h, T&... t) {
        bool f = read_single(h);
        assert(f);
        read(t...);
    }

    int read_unsafe() { return 0; }
    template <class H, class... T> int read_unsafe(H& h, T&... t) {
        bool f = read_single(h);
        if (!f) return 0;
        return 1 + read_unsafe(t...);
    }

    int close() { return ::close(fd); }

  private:
    static constexpr int SIZE = 1 << 15;

    int fd = -1;
    std::array<char, SIZE + 1> line;
    int st = 0, ed = 0;
    bool eof = false;

    bool read_single(std::string& ref) {
        if (!skip_space()) return false;
        ref = "";
        while (true) {
            char c = top();
            if (c <= ' ') break;
            ref += c;
            st++;
        }
        return true;
    }
    bool read_single(double& ref) {
        std::string s;
        if (!read_single(s)) return false;
        ref = std::stod(s);
        return true;
    }

    template <class T,
              std::enable_if_t<std::is_same<T, char>::value>* = nullptr>
    bool read_single(T& ref) {
        if (!skip_space<50>()) return false;
        ref = top();
        st++;
        return true;
    }

    template <class T,
              internal::is_signed_int_t<T>* = nullptr,
              std::enable_if_t<!std::is_same<T, char>::value>* = nullptr>
    bool read_single(T& sref) {
        using U = internal::to_unsigned_t<T>;
        if (!skip_space<50>()) return false;
        bool neg = false;
        if (line[st] == '-') {
            neg = true;
            st++;
        }
        U ref = 0;
        do {
            ref = 10 * ref + (line[st++] & 0x0f);
        } while (line[st] >= '0');
        sref = neg ? -ref : ref;
        return true;
    }
    template <class U,
              internal::is_unsigned_int_t<U>* = nullptr,
              std::enable_if_t<!std::is_same<U, char>::value>* = nullptr>
    bool read_single(U& ref) {
        if (!skip_space<50>()) return false;
        ref = 0;
        do {
            ref = 10 * ref + (line[st++] & 0x0f);
        } while (line[st] >= '0');
        return true;
    }

    bool reread() {
        if (ed - st >= 50) return true;
        if (st > SIZE / 2) {
            std::memmove(line.data(), line.data() + st, ed - st);
            ed -= st;
            st = 0;
        }
        if (eof) return false;
        auto u = ::read(fd, line.data() + ed, SIZE - ed);
        if (u == 0) {
            eof = true;
            line[ed] = '\0';
            u = 1;
        }
        ed += int(u);
        line[ed] = char(127);
        return true;
    }

    char top() {
        if (st == ed) {
            bool f = reread();
            assert(f);
        }
        return line[st];
    }

    template <int TOKEN_LEN = 0> bool skip_space() {
        while (true) {
            while (line[st] <= ' ') st++;
            if (ed - st > TOKEN_LEN) return true;
            if (st > ed) st = ed;
            for (auto i = st; i < ed; i++) {
                if (line[i] <= ' ') return true;
            }
            if (!reread()) return false;
        }
    }
};

struct Printer {
  public:
    template <char sep = ' ', bool F = false> void write() {}
    template <char sep = ' ', bool F = false, class H, class... T>
    void write(const H& h, const T&... t) {
        if (F) write_single(sep);
        write_single(h);
        write<true>(t...);
    }
    template <char sep = ' ', class... T> void writeln(const T&... t) {
        write<sep>(t...);
        write_single('\n');
    }

    Printer(FILE* _fp) : fd(fileno(_fp)) {}
    ~Printer() { flush(); }

    int close() {
        flush();
        return ::close(fd);
    }

    void flush() {
        if (pos) {
            auto res = ::write(fd, line.data(), pos);
            assert(res != -1);
            pos = 0;
        }
    }

  private:
    static std::array<std::array<char, 2>, 100> small;
    static std::array<unsigned long long, 20> tens;

    static constexpr size_t SIZE = 1 << 15;
    int fd;
    std::array<char, SIZE> line;
    size_t pos = 0;
    std::stringstream ss;

    template <class T,
              std::enable_if_t<std::is_same<char, T>::value>* = nullptr>
    void write_single(const T& val) {
        if (pos == SIZE) flush();
        line[pos++] = val;
    }

    template <class T,
              internal::is_signed_int_t<T>* = nullptr,
              std::enable_if_t<!std::is_same<char, T>::value>* = nullptr>
    void write_single(const T& val) {
        using U = internal::to_unsigned_t<T>;
        if (val == 0) {
            write_single('0');
            return;
        }
        if (pos > SIZE - 50) flush();
        U uval = val;
        if (val < 0) {
            write_single('-');
            uval = -uval;
        }
        write_unsigned(uval);
    }

    template <class U,
              internal::is_unsigned_int_t<U>* = nullptr,
              std::enable_if_t<!std::is_same<char, U>::value>* = nullptr>
    void write_single(U uval) {
        if (uval == 0) {
            write_single('0');
            return;
        }
        if (pos > SIZE - 50) flush();

        write_unsigned(uval);
    }

    static int calc_len(uint64_t x) {
        int i = ((63 - std::countl_zero(x)) * 3 + 3) / 10;
        if (x < tens[i])
            return i;
        else
            return i + 1;
    }

    template <class U,
              internal::is_unsigned_int_t<U>* = nullptr,
              std::enable_if_t<2 >= sizeof(U)>* = nullptr>
    void write_unsigned(U uval) {
        size_t len = calc_len(uval);
        pos += len;

        char* ptr = line.data() + pos;
        while (uval >= 100) {
            ptr -= 2;
            memcpy(ptr, small[uval % 100].data(), 2);
            uval /= 100;
        }
        if (uval >= 10) {
            memcpy(ptr - 2, small[uval].data(), 2);
        } else {
            *(ptr - 1) = char('0' + uval);
        }
    }

    template <class U,
              internal::is_unsigned_int_t<U>* = nullptr,
              std::enable_if_t<4 == sizeof(U)>* = nullptr>
    void write_unsigned(U uval) {
        std::array<char, 8> buf;
        memcpy(buf.data() + 6, small[uval % 100].data(), 2);
        memcpy(buf.data() + 4, small[uval / 100 % 100].data(), 2);
        memcpy(buf.data() + 2, small[uval / 10000 % 100].data(), 2);
        memcpy(buf.data() + 0, small[uval / 1000000 % 100].data(), 2);

        if (uval >= 100000000) {
            if (uval >= 1000000000) {
                memcpy(line.data() + pos, small[uval / 100000000 % 100].data(),
                       2);
                pos += 2;
            } else {
                line[pos] = char('0' + uval / 100000000);
                pos++;
            }
            memcpy(line.data() + pos, buf.data(), 8);
            pos += 8;
        } else {
            size_t len = calc_len(uval);
            memcpy(line.data() + pos, buf.data() + (8 - len), len);
            pos += len;
        }
    }

    template <class U,
              internal::is_unsigned_int_t<U>* = nullptr,
              std::enable_if_t<8 == sizeof(U)>* = nullptr>
    void write_unsigned(U uval) {
        size_t len = calc_len(uval);
        pos += len;

        char* ptr = line.data() + pos;
        while (uval >= 100) {
            ptr -= 2;
            memcpy(ptr, small[uval % 100].data(), 2);
            uval /= 100;
        }
        if (uval >= 10) {
            memcpy(ptr - 2, small[uval].data(), 2);
        } else {
            *(ptr - 1) = char('0' + uval);
        }
    }

    template <
        class U,
        std::enable_if_t<internal::is_unsigned_int128<U>::value>* = nullptr>
    void write_unsigned(U uval) {
        static std::array<char, 50> buf;
        size_t len = 0;
        while (uval > 0) {
            buf[len++] = char((uval % 10) + '0');
            uval /= 10;
        }
        std::reverse(buf.begin(), buf.begin() + len);
        memcpy(line.data() + pos, buf.data(), len);
        pos += len;
    }

    void write_single(const std::string& s) {
        for (char c : s) write_single(c);
    }
    void write_single(const char* s) {
        size_t len = strlen(s);
        for (size_t i = 0; i < len; i++) write_single(s[i]);
    }
    template <class T> void write_single(const std::vector<T>& val) {
        auto n = val.size();
        for (size_t i = 0; i < n; i++) {
            if (i) write_single(' ');
            write_single(val[i]);
        }
    }
};

inline std::array<std::array<char, 2>, 100> Printer::small = [] {
    std::array<std::array<char, 2>, 100> table;
    for (int i = 0; i <= 99; i++) {
        table[i][1] = char('0' + (i % 10));
        table[i][0] = char('0' + (i / 10 % 10));
    }
    return table;
}();
inline std::array<unsigned long long, 20> Printer::tens = [] {
    std::array<unsigned long long, 20> table;
    for (int i = 0; i < 20; i++) {
        table[i] = 1;
        for (int j = 0; j < i; j++) {
            table[i] *= 10;
        }
    }
    return table;
}();

}  // namespace yosupo
#line 2 "/home/vscode/yosupo-library/src/yosupo/hashmap.hpp"

#include <iterator>
#include <utility>
#line 6 "/home/vscode/yosupo-library/src/yosupo/hashmap.hpp"

#line 2 "/home/vscode/yosupo-library/src/yosupo/hash.hpp"

#line 5 "/home/vscode/yosupo-library/src/yosupo/hash.hpp"
#include <tuple>
#line 7 "/home/vscode/yosupo-library/src/yosupo/hash.hpp"
#include <set>
#include <map>

#line 2 "/home/vscode/yosupo-library/src/yosupo/random.hpp"

#line 6 "/home/vscode/yosupo-library/src/yosupo/random.hpp"
#include <chrono>
#line 8 "/home/vscode/yosupo-library/src/yosupo/random.hpp"
#include <random>
#line 10 "/home/vscode/yosupo-library/src/yosupo/random.hpp"

namespace yosupo {

struct Xoshiro256StarStar {
  public:
    using result_type = uint64_t;
    Xoshiro256StarStar() : Xoshiro256StarStar(0) {}
    explicit Xoshiro256StarStar(uint64_t seed) {
        // use splitmix64
        // Reference: http://xoshiro.di.unimi.it/splitmix64.c
        for (int i = 0; i < 4; i++) {
            uint64_t z = (seed += 0x9e3779b97f4a7c15);
            z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
            z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
            s[i] = z ^ (z >> 31);
        }
    }

    static constexpr result_type min() { return 0; }
    static constexpr result_type max() { return -1; }

    result_type operator()() {
        const uint64_t result_starstar = rotl(s[1] * 5, 7) * 9;

        const uint64_t t = s[1] << 17;

        s[2] ^= s[0];
        s[3] ^= s[1];
        s[1] ^= s[2];
        s[0] ^= s[3];

        s[2] ^= t;

        s[3] = rotl(s[3], 45);

        return result_starstar;
    }

  private:
    // Use xoshiro256**
    // Refereces: http://xoshiro.di.unimi.it/xoshiro256starstar.c
    static uint64_t rotl(const uint64_t x, int k) {
        return (x << k) | (x >> (64 - k));
    }

    std::array<uint64_t, 4> s;
};

// https://github.com/wangyi-fudan/wyhash
struct WYRand {
  public:
    using result_type = uint64_t;
    explicit WYRand(uint64_t seed) : s(seed) {}

    static constexpr result_type min() { return 0; }
    static constexpr result_type max() { return -1; }

    result_type operator()() {
        s += 0x2d358dccaa6c78a5;
        auto x = (unsigned __int128)s * (s ^ 0x8bb84b93962eacc9);
        return (uint64_t)(x ^ (x >> 64));
    }

  private:
    uint64_t s;
};
using Random = WYRand;
inline Random& global_gen() {
    static Random gen(
        std::chrono::steady_clock::now().time_since_epoch().count());
    return gen;
}

template <class G>
concept random_64 = std::uniform_random_bit_generator<G> &&
                    std::same_as<uint64_t, std::invoke_result_t<G&>> &&
                    G::min() == uint64_t(0) && G::max() == uint64_t(-1);

namespace internal {

// random choice from [0, upper]
template <random_64 G> uint64_t uniform_u64(uint64_t upper, G& gen) {
    if (!(upper & (upper + 1))) {
        // b = 00..0011..11
        return gen() & upper;
    }
    int log = 63 - std::countl_zero(upper);
    uint64_t mask = (log == 63) ? ~0ULL : (1ULL << (log + 1)) - 1;
    while (true) {
        uint64_t r = gen() & mask;
        if (r <= upper) return r;
    }
}

// random choice from [0, upper], faster than uniform_u64
template <random_64 G> uint64_t random_u64(uint64_t upper, G& gen) {
    return (uint64_t)(((unsigned __int128)(upper) + 1) * gen() >> 64);
}

}  // namespace internal

template <class T, random_64 G> T uniform(T lower, T upper, G& gen) {
    return T(lower +
             internal::uniform_u64(uint64_t(upper) - uint64_t(lower), gen));
}
template <class T> T uniform(T lower, T upper) {
    return uniform(lower, upper, global_gen());
}
template <class T, random_64 G> T random(T lower, T upper, G& gen) {
    return T(lower +
             internal::random_u64(uint64_t(upper) - uint64_t(lower), gen));
}
template <class T> T random(T lower, T upper) {
    return random(lower, upper, global_gen());
}

template <random_64 G> bool uniform_bool(G& gen) { return gen() & 1; }
inline bool uniform_bool() { return uniform_bool(global_gen()); }

// select 2 elements from [lower, uppper]
template <class T, random_64 G>
std::pair<T, T> uniform_pair(T lower, T upper, G& gen) {
    assert(upper - lower >= 1);
    T a, b;
    do {
        a = uniform(lower, upper, gen);
        b = uniform(lower, upper, gen);
    } while (a == b);
    if (a > b) std::swap(a, b);
    return {a, b};
}
template <class T> std::pair<T, T> uniform_pair(T lower, T upper) {
    return uniform_pair(lower, upper, global_gen());
}

// random value in the interval (0.0, 1.0]
template <class G> inline double open_closed_01(G& gen) {
    union {
        uint64_t i;
        double f;
    } u = {0xbff0000000000000 | (gen() >> 12)};
    return 2.0 + u.f;
}
inline double open_closed_01() {
    return open_closed_01(global_gen());
}

}  // namespace yosupo
#line 2 "/home/vscode/yosupo-library/src/yosupo/bitvector.hpp"

#line 7 "/home/vscode/yosupo-library/src/yosupo/bitvector.hpp"

namespace yosupo {

struct BitVec {
  public:
    BitVec() : BitVec(0) {}
    explicit BitVec(size_t _n) : n(_n), d((n + B - 1) / B) {}
    size_t size() const { return n; }

    void reset() { fill(d.begin(), d.end(), 0); }
    void reset(size_t i) { set(i, false); }
    BitVec& set() {
        for (auto& x : d) {
            x = -1ULL;
        }
        erase_last();
        return *this;
    }
    BitVec& set(size_t i, bool f = true) {
        assert(i < n);
        if (f) {
            d[i / B] |= 1ULL << (i % B);
        } else {
            d[i / B] &= ~(1ULL << (i % B));
        }
        return *this;
    }
    bool test(size_t i) const {
        assert(i < n);
        return (d[i / B] >> (i % B) & 1) != 0;
    }

    size_t count() const {
        size_t sm = 0;
        for (auto x : d) sm += std::popcount(x);
        return sm;
    }
    bool any() const {
        for (auto& x : d)
            if (x) return true;
        return false;
    }
    int find_first() const {
        auto m = d.size();
        for (size_t i = 0; i < m; i++) {
            if (d[i]) return int(i * B + std::countr_zero(d[i]));
        }
        return int(size());
    }

    BitVec& flip() {
        op1(std::bit_not<unsigned long long>());
        if (n % B) d.back() &= ~(-1ULL << (n % B));
        return *this;
    }
    BitVec& flip(size_t i) {
        return set(i, !test(i));
    }

    // operators
    BitVec& operator~() const { return BitVec(*this).flip(); }
    BitVec& operator&=(const BitVec& r) {
        return op2(r, std::bit_and<unsigned long long>());
    }
    BitVec& operator|=(const BitVec& r) {
        return op2(r, std::bit_or<unsigned long long>());
    }
    BitVec& operator^=(const BitVec& r) {
        return op2(r, std::bit_xor<unsigned long long>());
    }
    BitVec& operator<<=(const size_t& s) {
        auto block = s / B, rem = s % B;
        if (d.size() <= block) {
            reset();
            return *this;
        }
        for (size_t i = d.size() - 1; i > block; i--) {
            if (rem == 0)
                d[i] = d[i - block];
            else
                d[i] = d[i - block] << rem | d[i - block - 1] >> (B - rem);
        }
        d[block] = d[0] << rem;
        erase_last();
        fill(d.begin(), d.begin() + block, 0ULL);
        return *this;
    }
    BitVec& operator>>=(const size_t& s) {
        auto block = s / B, rem = s % B;
        if (d.size() <= block) {
            reset();
            return *this;
        }
        for (size_t i = 0; i < d.size() - block - 1; i++) {
            if (rem == 0)
                d[i] = d[i + block];
            else
                d[i] = d[i + block + 1] << (B - rem) | d[i + block] >> rem;
        }
        d[d.size() - block - 1] = d.back() >> rem;
        fill(d.begin() + d.size() - block, d.end(), 0ULL);
        return *this;
    }
    BitVec operator&(const BitVec& r) const { return BitVec(*this) &= r; }
    BitVec operator|(const BitVec& r) const { return BitVec(*this) |= r; }
    BitVec operator^(const BitVec& r) const { return BitVec(*this) ^= r; }
    BitVec operator<<(const size_t& s) const { return BitVec(*this) <<= s; }
    BitVec operator>>(const size_t& s) const { return BitVec(*this) >>= s; }
    bool operator==(const BitVec& r) const { return d == r.d; }
    bool operator!=(const BitVec& r) const { return !(d == r.d); }

    void push_back(bool f) {
        if (n % B == 0) d.push_back(0);
        set(n, f);
        n++;
    }

    BitVec substr(size_t st, size_t le) const {
        assert(st + le <= n);
        BitVec res(le);
        size_t i = 0;
        while (i < le) {
            res.d[i / B] |= d[(st + i) / B] >> ((st + i) % B) << (i % B);
            i += std::min(B - i % B, B - (st + i) % B);
        }
        res.erase_last();
        return res;
    }
    bool substr_match(size_t st, const BitVec& pat) const {
        size_t le = pat.size();
        assert(st + le <= n);
        size_t i = 0;
        while (i < le) {
            size_t u = std::min({le - i, B - i % B, B - (st + i) % B});
            unsigned long long z = pat.d[i / B] >> (i % B) ^ d[(st + i) / B] >> ((st + i) % B);
            if (z << (B - u)) return false;
            i += u;
        }
        return true;
    }

    const std::vector<unsigned long long>& raw_data() const { return d; }

  private:
    static constexpr size_t B = 64;
    size_t n;
    std::vector<unsigned long long> d;
    void erase_last() {
        if (n % B) d.back() &= (unsigned long long)(-1) >> (B - n % B);
    }

    template <class OP> BitVec& op1(OP op) {
        for (auto& x : d) x = op(x);
        return *this;
    }

    template <class OP> BitVec& op2(const BitVec& r, OP op) {
        assert(n == r.n);
        for (size_t i = 0; i < d.size(); i++) d[i] = op(d[i], r.d[i]);
        return *this;
    }
};

namespace internal {

// bitvec
template <class H> auto update(const H& h, const BitVec& bs) {
    return update(h, bs.raw_data());
}

}  // namespace internal

}  // namespace yosupo
#line 13 "/home/vscode/yosupo-library/src/yosupo/hash.hpp"

namespace yosupo {

namespace internal {

constexpr int THRESHOLD_HASH = 100;

// Reference: Lemire, Daniel., and Owen, Kaser.
// Strongly Universal String Hashing Is Fast.
inline uint64_t get_seed(int i) {
    static std::array<uint64_t, THRESHOLD_HASH> seed = []() {
        std::array<uint64_t, THRESHOLD_HASH> _seed;
        for (auto& x : _seed) {
            x = yosupo::uniform(uint64_t(0), uint64_t(-1));
        }
        return _seed;
    }();
    static std::vector<uint64_t> long_seed;

    if (i < THRESHOLD_HASH) return seed[i];

    while (THRESHOLD_HASH + int(long_seed.size()) <= i) {
        long_seed.push_back(yosupo::uniform(uint64_t(0), uint64_t(-1)));
    }

    return long_seed[i - THRESHOLD_HASH];
}

struct DynHasher32 {
    int len = 0;
    uint64_t v = get_seed(0);

    DynHasher32 update32(uint32_t x) const {
        return DynHasher32{len + 1, v + x * get_seed(len)};
    }

    DynHasher32 to_dyn() const { return *this; }
    uint32_t digest() const { return uint32_t(v >> 32); }
};

template <int I = 1> struct Hasher32 {
    uint64_t v = get_seed(0);

    Hasher32<I + 1> update32(uint32_t x) const {
        return Hasher32<I + 1>{v + x * get_seed(I)};
    }

    DynHasher32 to_dyn() const { return DynHasher32{I, v}; }
    uint32_t digest() const { return uint32_t(v >> 32); }
};

// integer
template <class H,
          class T,
          is_integral_t<T>* = nullptr,
          std::enable_if_t<4 >= sizeof(T)>* = nullptr>
auto update(const H& h, const T& x) {
    return h.update32(uint32_t(x));
}
template <class H,
          class T,
          is_integral_t<T>* = nullptr,
          std::enable_if_t<sizeof(T) == 8>* = nullptr>
auto update(const H& h, const T& x) {
    return update(update(h, uint32_t(x)), uint32_t(uint64_t(x) >> 32));
}
template <class H,
          class T,
          is_integral_t<T>* = nullptr,
          std::enable_if_t<sizeof(T) == 16>* = nullptr>
auto update(const H& h, const T& x) {
    return update(update(h, uint64_t(x)), uint64_t((__uint128_t)(x) >> 64));
}

// pair
template <class H, class S, class T>
auto update(const H& h, const std::pair<S, T>& x) {
    return update(update(h, x.first), x.second);
}

// tuple
template <int I,
          class H,
          class T,
          std::enable_if_t<(I == std::tuple_size<T>::value)>* = nullptr>
auto update(const H& h, const T&) {
    return h;
}
template <int I = 0,
          class H,
          class T,
          std::enable_if_t<(I != std::tuple_size<T>::value)>* = nullptr>
auto update(const H& h, const T& x) {
    return update<I + 1>(update(h, std::get<I>(x)), x);
}

// array
template <int I = 0,
          class H,
          class T,
          int N>
auto update(const H& h, const std::array<T, N>& x) {
    return I == N ? h : update<I + 1>(update(h, x[I]), x);
}

// vector
template <class H, class T> auto update(const H& h, const std::vector<T>& v) {
    auto h2 = update(h, v.size()).to_dyn();
    for (const auto& x : v) {
        h2 = update(h2, x);
    }
    return h2;
}

// set
template <class H, class T> auto update(const H& h, const std::set<T>& s) {
    auto h2 = update(h, s.size()).to_dyn();
    for (const auto& x : s) {
        h2 = update(h2, x);
    }
    return h2;
}

// map
template <class H, class T, class U> auto update(const H& h, const std::map<T, U>& m) {
    auto h2 = update(h, m.size()).to_dyn();
    for (const auto& x : m) {
        h2 = update(h2, x);
    }
    return h2;
}

}  // namespace internal

template <class T> struct UniversalHash32 {
    uint32_t operator()(const T& x) {
        return internal::update(internal::Hasher32<>{}, x).digest();
    }
};

}  // namespace yosupo
#line 8 "/home/vscode/yosupo-library/src/yosupo/hashmap.hpp"

namespace yosupo {

template <class K, class D, class H = UniversalHash32<K>>
struct IncrementalHashMap {
  public:
    using Data = std::pair<K, D>;

  private:
    struct Iterator {
      public:
        using difference_type = int;
        using value_type = Data;
        using pointer = Data*;
        using reference = Data&;
        using iterator_category = std::forward_iterator_tag;

        IncrementalHashMap& _mp;
        unsigned int _pos;
        Iterator(IncrementalHashMap& mp, unsigned int pos)
            : _mp(mp), _pos(pos) {}

        std::pair<K, D>& operator*() const { return _mp.data[_pos]; }
        std::pair<K, D>* operator->() const { return &_mp.data[_pos]; }

        Iterator& operator++() {
            _pos = _mp.next_bucket(_pos + 1);
            return *this;
        }
        Iterator operator++(int) {
            auto result = *this;
            ++*this;
            return result;
        }

        bool operator==(const Iterator& rhs) const { return _pos == rhs._pos; }
        bool operator!=(const Iterator& rhs) const { return !(*this == rhs); }
    };

  public:
    IncrementalHashMap(size_t s)
        : mask((1 << s) - 1), filled(0), used(mask + 1), data(mask + 1) {}
    IncrementalHashMap() : IncrementalHashMap(2) {}

    Iterator begin() { return Iterator(*this, next_bucket(0)); }
    Iterator end() { return Iterator(*this, mask + 1); }
    using iterator = Iterator;

    D& operator[](const K& k) {
        unsigned int i = H()(k) & mask;
        while (used[i] && data[i].first != k) {
            i = (i + 1) & mask;
        }
        if (!used[i]) {
            if (filled * 2 > mask) {
                rehash();
                return (*this)[k];
            }
            filled++;
            used[i] = true;
            data[i] = {k, D()};
        }
        return data[i].second;
    }

    Iterator find(const K& k) {
        unsigned int i = H()(k) & mask;
        while (used[i] && data[i].first != k) {
            i = (i + 1) & mask;
        }
        if (!used[i]) return end();
        return Iterator(*this, i);
    }

    size_t count(const K& k) {
        return find(k) == end() ? 0 : 1;
    }

    size_t size() const { return filled; }

  private:
    unsigned int mask, filled;  // data.size() == 1 << s

    std::vector<bool> used;
    std::vector<Data> data;

    void rehash() {
        unsigned int pmask = mask;
        mask = mask * 2 + 1;
        filled = 0;
        auto pused = std::exchange(used, std::vector<bool>(mask + 1));
        auto pdata = std::exchange(data, std::vector<Data>(mask + 1));
        for (unsigned int i = 0; i <= pmask; i++) {
            if (pused[i]) {
                (*this)[pdata[i].first] = pdata[i].second;
            }
        }
    }

    unsigned int next_bucket(unsigned int i) const {
        while (i <= mask && !used[i]) i++;
        return i;
    }
};

}  // namespace yosupo
#line 6 "ucup3-16/F/main.cpp"
using namespace yosupo;

#line 2 "ucup3-16/F/base.hpp"

#ifdef YOSUPO_AVX2_PRAGMA
#line 5 "ucup3-16/F/base.hpp"
#pragma GCC target("avx2,bmi,bmi2,lzcnt,popcnt")
#endif

#line 11 "ucup3-16/F/base.hpp"
#include <bitset>
#line 13 "ucup3-16/F/base.hpp"
#include <cmath>
#include <cstdio>
#line 16 "ucup3-16/F/base.hpp"
#include <iostream>
#line 18 "ucup3-16/F/base.hpp"
#include <queue>
#include <ranges>
#line 24 "ucup3-16/F/base.hpp"

using std::abs, std::pow, std::sqrt;
using std::array, std::vector, std::string, std::queue, std::deque;
using std::countl_zero, std::countl_one, std::countr_zero, std::countr_one;
using std::istream, std::ostream, std::cerr, std::endl;
using std::min, std::max, std::swap;
using std::pair, std::tuple, std::bitset;
using std::popcount;
using std::priority_queue, std::set, std::multiset, std::map;
using std::views::iota, std::views::reverse;

namespace ranges = std::ranges;
using ranges::sort, ranges::copy_n;

using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T> using V = vector<T>;
template <class T> using VV = V<V<T>>;

#ifdef YOSUPO_LOCAL

inline ostream& operator<<(ostream& os, __int128_t x) {
    if (x < 0) {
        os << "-";
        x *= -1;
    }
    if (x == 0) {
        return os << "0";
    }
    string s;
    while (x) {
        s += char(x % 10 + '0');
        x /= 10;
    }
    ranges::reverse(s);
    return os << s;
}
inline ostream& operator<<(ostream& os, __uint128_t x) {
    if (x == 0) {
        return os << "0";
    }
    string s;
    while (x) {
        s += char(x % 10 + '0');
        x /= 10;
    }
    ranges::reverse(s);
    return os << s;
}

template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p);
template <class T> ostream& operator<<(ostream& os, const V<T>& v);
template <class T> ostream& operator<<(ostream& os, const deque<T>& v);
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a);
template <class T> ostream& operator<<(ostream& os, const set<T>& s);
template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& m);

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

template <class T> ostream& operator<<(ostream& os, const V<T>& v) {
    os << "[";
    bool f = false;
    for (auto d : v) {
        if (f) os << ", ";
        f = true;
        os << d;
    }
    return os << "]";
}

template <class T> ostream& operator<<(ostream& os, const deque<T>& v) {
    os << "[";
    bool f = false;
    for (auto d : v) {
        if (f) os << ", ";
        f = true;
        os << d;
    }
    return os << "]";
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a) {
    os << "[";
    bool f = false;
    for (auto d : a) {
        if (f) os << ", ";
        f = true;
        os << d;
    }
    return os << "]";
}

template <class T> ostream& operator<<(ostream& os, const set<T>& s) {
    os << "{";
    bool f = false;
    for (auto d : s) {
        if (f) os << ", ";
        f = true;
        os << d;
    }
    return os << "}";
}
template <class T> ostream& operator<<(ostream& os, const multiset<T>& s) {
    os << "{";
    bool f = false;
    for (auto d : s) {
        if (f) os << ", ";
        f = true;
        os << d;
    }
    return os << "}";
}

template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& s) {
    os << "{";
    bool f = false;
    for (auto p : s) {
        if (f) os << ", ";
        f = true;
        os << p.first << ": " << p.second;
    }
    return os << "}";
}

struct PrettyOS {
    ostream& os;
    bool first;

    template <class T> auto operator<<(T&& x) {
        if (!first) os << ", ";
        first = false;
        os << x;
        return *this;
    }
};
template <class... T> void dbg0(T&&... t) {
    (PrettyOS{cerr, true} << ... << t);
}
#define dbg(...)                                            \
    do {                                                    \
        cerr << __LINE__ << " : " << #__VA_ARGS__ << " = "; \
        dbg0(__VA_ARGS__);                                  \
        cerr << endl;                                       \
    } while (false);
#else
#define dbg(...)
#endif
#line 9 "ucup3-16/F/main.cpp"

Scanner sc = Scanner(stdin);
Printer pr = Printer(stdout);

namespace noimi {
#define pii pair<int, int>
#define pll pair<ll, ll>
#define vi vector<int>
#define vl vector<ll>
#define ov3(a, b, c, name, ...) name
#define rep0(n) for (ll aaaaa = 0; aaaaa < n; ++aaaaa)
#define rep1(i, n) for (ll i = 0; i < (n); i++)
#define rep2(i, a, b) for (ll i = (a); i < (b); i++)
#define rep(...) ov3(__VA_ARGS__, rep2, rep1, rep0)(__VA_ARGS__)
#define per(i, a, b) for (ll i = (a) - 1; i >= (b); i--)
#define all(a) begin(a), end(a)
#define si(a) (int)(size(a))
#define lb(v, x) (lower_bound(all(v), x) - begin(v))
#define eb emplace_back
bool chmin(auto& a, auto b) { return a > b ? a = b, 1 : 0; }
bool chmax(auto& a, auto b) { return a < b ? a = b, 1 : 0; }

const int INF = int(1e9 + 100);
const ll INFL = ll(3e18 + 100);

struct lctree {
    struct line {
        ll a, b;
        line() : a(0), b(INFL) {}
        line(ll a, ll b) : a(a), b(b) {}
        ll get(ll x) { return a * x + b; }
        inline bool over(line r, ll x) { return get(x) < r.get(x); }
    };
    int n;

    vector<ll> x;
    vector<line> seg;
    lctree() {}
    lctree(const vector<ll>& _x) : x(_x) {
        sort(all(x));
        int n2 = si(x);
        n = 1;
        while (n < n2) n <<= 1;
        x.resize(n);
        rep(i, n2, n) x[i] = x[n2 - 1];
        seg = vector<line>(n * 2);
    }
    void upd(line L, int i, int l, int r) {
        while (true) {
            int mid = l + r >> 1;
            bool lov = L.over(seg[i], x[l]);
            bool rov = L.over(seg[i], x[r - 1]);
            if (lov == rov) {
                if (lov) swap(seg[i], L);
                return;
            }
            bool mov = L.over(seg[i], x[mid]);
            if (mov) swap(seg[i], L);
            if (lov != mov) {
                i = (i << 1), r = mid;
            } else {
                i = (i << 1) + 1, l = mid;
            }
        }
    }
    void upd(line L, unsigned i) {
        int ub = std::bit_width(i) - 1;
        int l = (n >> ub) * (i - (1 << ub));
        int r = l + (n >> ub);
        upd(L, i, l, r);
    }
    void update(ll a, ll b) { upd(line(a, b), 1, 0, n); }
    void update_segment(ll l, ll r, ll a, ll b) {
        l = lb(x, l) + n, r = lb(x, r) + n;
        line L(a, b);
        for (; l < r; l >>= 1, r >>= 1) {
            if (l & 1) upd(L, l++);
            if (r & 1) upd(L, --r);
        }
    }
    ll query(ll t) {
        ll k = lb(x, t);
        k += n;
        ll res = seg[k].get(t);
        while (k > 1) {
            k >>= 1;
            chmin(res, seg[k].get(t));
        }
        return res;
    }
};

}

int main() {
    int n, lines;
    sc.read(n, lines);

    V<ll> a(lines), b(lines);
    for (int i : iota(0, lines)) {
        sc.read(a[i]);
    }
    for (int i : iota(0, lines)) {
        sc.read(b[i]);
    }

    using P = pair<int, int>;
    VV<P> ps(n); // line, lid

    VV<int> vs(lines);
    VV<ll> ws(lines);
    for (int i : iota(0, lines)) {
        int k;
        sc.read(k);

        vs[i].resize(k);
        ws[i].resize(k - 1);
        sc.read(vs[i][0]);
        vs[i][0]--;
        for (int j : iota(0, k - 1)) {
            sc.read(ws[i][j], vs[i][j + 1]);
            vs[i][j + 1]--;
        }

        for (int j : iota(0, k)) {
            ps[vs[i][j]].push_back({i, j});
        }
    }

    for (int i : iota(0, n)) {
        sort(ps[i].begin(), ps[i].end(), [&](P x, P y) {
            return a[x.first] < a[y.first];
        });
    }

    dbg(ps);

    IncrementalHashMap<P, int> vl2vid;
    for (int i : iota(0, n)) {
        for (int j : iota(0, int(ps[i].size()))) {
            vl2vid[{i, ps[i][j].first}] = j;
        }
    }

    using S = pair<bool, P>; // type, (v, vid)
    using Q = pair<ll, S>;
    priority_queue<Q, V<Q>, std::greater<Q>> que;

    const ll INF = TEN(18) + TEN(10);
    VV<ll> dist(n);
    for (int i : iota(0, n)) {
        dist[i].assign(ps[i].size(), INF);        
    }
    V<int> cht(n);

    V<ll> cht_dist(n, INF);

    V<noimi::lctree> chts;
    for (int i : iota(0, n)) {
        V<ll> x;
        for (int j : iota(0, int(ps[i].size()))) {
            x.push_back(b[ps[i][j].first]);
        }
        chts.push_back(noimi::lctree(x));
    }

    auto check = [&](int v) {
        if (cht[v] == int(ps[v].size())) return;

        int to = ps[v][cht[v]].first;

        // // SLOW
        // ll mi = INF;
        // for (int i : iota(0, int(ps[v].size()))) {
        //     int from = ps[v][i].first;
        //     mi = min(mi, dist[v][i] + b[from] * a[to]);
        // }

        ll mi = chts[v].query(a[to]);
        if (cht_dist[v] <= mi) return;
        cht_dist[v] = mi;
        que.push({mi, {1, {v, cht[v]}}});
    };

    auto upd = [&](P p, ll di) { // p = (v, vid)
        if (dist[p.first][p.second] <= di) return;
        dist[p.first][p.second] = di;
        chts[p.first].update(b[ps[p.first][p.second].first], di);

        que.push({di, {0, p}});

        check(p.first);
    };
    for (int i : iota(0, int(ps[0].size()))) {
        upd({0, i}, 0);
    }

    V<ll> ans(n, INF);    
    while (que.size()) {
        auto [di, s] = que.top();
        auto [type, p] = s;
        auto [v, id] = p;
        que.pop();
        ans[v] = min(ans[v], di);

        if (type == 1) {
            if (id != cht[v]) continue;
            cht[v]++;
            cht_dist[v] = INF;
            upd(p, di);
            check(v);
        } else {
            // station
            auto [line, index] = ps[v][id];
            if (index + 1 < int(vs[line].size())) {
                int nv = vs[line][index + 1];
                int nvid = vl2vid[{nv, line}];
                upd({nv, nvid}, di + ws[line][index]);
            }
        }
    }

    for (int i : iota(1, n)) {
        if (i > 1) pr.write(' ');
        pr.write(ans[i]);
    }
    pr.writeln();
    return 0;
}