#ifdef DEBUG
    #define _GLIBCXX_DEBUG
#endif
#pragma GCC optimize("Ofast")
#include <cassert>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <functional>
#include <stack>
#include <queue>
#include <deque>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <complex>
#include <iomanip>
#include <bitset>
#include <random>
#include <chrono>

#define stdin_filepath "CON"
#define stdout_filepath "stdout.txt"
#define stderr_filepath "stderr.txt"
#define debug_stream std::cerr
#define iostream_untie true
#define __precision__ 10

#define rep(i,n) for(int i = 0; i < int(n); ++i)
#define all(v) begin(v), end(v)
#define rall(v) rbegin(v), rend(v)
#define mkp make_pair
#define mkt make_tuple
#define popcnt __builtin_popcountll

using namespace std;
using i64 = int_fast64_t;
using pii = std::pair<int, int>;
using pll = std::pair<int_fast64_t, int_fast64_t>;
template <class T> using heap = std::priority_queue<T>;
template <class T> using minheap = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template <class T> constexpr T inf = std::numeric_limits<T>::max() / (T)2 - (T)1123456;
constexpr int dx[9] = {1, 0, -1, 0, 1, -1, -1, 1, 0};
constexpr int dy[9] = {0, 1, 0, -1, 1, 1, -1, -1, 0};

namespace execution
{
    std::chrono::system_clock::time_point start_time, end_time;
    void print_elapsed_time()
    {
        end_time = std::chrono::system_clock::now();
        std::cerr << "\n----- Exec time : ";
        std::cerr << std::chrono::duration_cast<std::chrono::milliseconds>(end_time - start_time).count();
        std::cerr << " ms -----\n";
    }
    struct setupper
    {
        setupper()
        {
            if(iostream_untie)
            {
                std::ios::sync_with_stdio(false);
                std::cin.tie(nullptr);
                // std::cout.tie(nullptr);
                // std::cerr.tie(nullptr);
            }
            std::cout << std::fixed << std::setprecision(__precision__);
            std::cerr << std::fixed << std::setprecision(__precision__);
    #ifdef LOCAL
            if(!freopen(stdout_filepath, "wt", stdout))
            {
                freopen("CON", "wt", stdout);
                std::cerr << "Failed to open the stdout file\n\n";
            }
            if(!freopen(stdin_filepath, "rt", stdin))
            {
                freopen("CON", "rt", stdin);
                std::cerr << "Failed to open the stdin file\n\n";
            }
            if(!freopen(stderr_filepath, "wt", stderr))
            {
                freopen("CON", "wt", stderr);
                std::cerr << "Failed to open the stderr file\n\n";
            }
            std::cout << "", std::cerr << "";
    #endif
    #if defined(LOCAL) || defined(DEBUG)
            atexit(print_elapsed_time);
            start_time = std::chrono::system_clock::now();
    #endif
        }
    } __setupper;
}

struct myclock_t
{
    std::chrono::system_clock::time_point built_pt, last_pt;
    int built_ln, last_ln;
    std::string built_func, last_func;
    bool is_built;
    myclock_t() : is_built(false) {}
    void build(int crt_ln, const std::string &crt_func)
    {
        is_built = true;
        last_pt = built_pt = std::chrono::system_clock::now();
        last_ln = built_ln = crt_ln, last_func = built_func = crt_func;
    }
    void set(int crt_ln, const std::string &crt_func)
    {
        if(is_built)
        {
            last_pt = std::chrono::system_clock::now();
            last_ln = crt_ln, last_func = crt_func;
        }
        else
        {
            debug_stream << "[ " << crt_ln << " : " << crt_func << " ] " << "myclock_t::set failed (yet to be built clock!)\n";
        }
    }
    void get(int crt_ln, const std::string &crt_func) {
        if(is_built) 
        {
            std::chrono::system_clock::time_point crt_pt(std::chrono::system_clock::now());
            int64_t diff = std::chrono::duration_cast<std::chrono::milliseconds>(crt_pt - last_pt).count();
            debug_stream << diff << " ms elapsed from" << " [ " <<  last_ln << " : " << last_func << " ]";
            if(last_ln == built_ln) debug_stream << " (when built)";
            debug_stream << " to" << " [ " << crt_ln << " : " << crt_func << " ]" << "\n";
            last_pt = built_pt, last_ln = built_ln, last_func = built_func;
        }
        else
        {
            debug_stream << "[ " << crt_ln << " : " << crt_func << " ] " << "myclock_t::get failed (yet to be built clock!)\n";
        }
    }
};
#if defined(LOCAL) || defined(DEBUG)
    myclock_t myclock;
    #define build_clock() myclock.build(__LINE__, __func__)
    #define set_clock() myclock.set(__LINE__, __func__)
    #define get_clock() myclock.get(__LINE__, __func__)
#else
    #define build_clock()
    #define set_clock()
    #define get_clock()
#endif

namespace std
{
    template <class RAitr> void rsort(RAitr __first, RAitr __last) { sort(__first, __last, greater<>()); }
    template <class T> size_t hash_combine(size_t seed, T const &key) { return seed ^ (hash<T>()(key) + 0x9e3779b9 + (seed << 6) + (seed >> 2)); }
    template <class T, class U> struct hash<pair<T,U>> { size_t operator()(pair<T,U> const &pr) const { return hash_combine(hash_combine(0, pr.first), pr.second); } };
    template <class tuple_t, size_t index = tuple_size<tuple_t>::value - 1>
    struct tuple_hash_calc { static size_t apply(size_t seed, tuple_t const &t) { return hash_combine(tuple_hash_calc<tuple_t, index - 1>::apply(seed, t), get<index>(t)); } };
    template <class tuple_t> struct tuple_hash_calc<tuple_t, 0> { static size_t apply(size_t seed, tuple_t const &t) { return hash_combine(seed, get<0>(t)); } };
    template <class ...T> struct hash<tuple<T...>> { size_t operator()(tuple<T...> const &t) const { return tuple_hash_calc<tuple<T...>>::apply(0, t); } };
    template <class T, class U> istream &operator>> (std::istream &s, pair<T,U> &p) { return s >> p.first >> p.second; }
    template <class T, class U> ostream &operator<< (std::ostream &s, const pair<T,U> p) { return s << p.first << " " << p.second; }
    template <class T> istream &operator>> (istream &s, vector<T> &v) { for(T &e : v) { s >> e; } return s; }
    template <class T> ostream &operator<< (ostream &s, const vector<T> &v) { for(size_t i = 0; i < v.size(); ++i) { s << (i ? " " : "") << v[i]; } return s; }
    template <class tuple_t, size_t index>
    struct tupleos
    {
        static ostream &apply(ostream &s, const tuple_t &t)
        {
            tupleos<tuple_t,index - 1>::apply(s,t);
            return s << " " << get<index>(t);
        }
    };
    template <class tuple_t>
    struct tupleos<tuple_t, 0>
    {
        static ostream &apply(ostream &s, const tuple_t &t)
        {
            return s << get<0>(t);
        }
    };
    template <class ...T> ostream &operator<< (ostream &s, const tuple<T...> &t)
    {
        return tupleos<tuple<T...>, tuple_size<tuple<T...>>::value - 1>::apply(s,t);
    }
    template <> ostream &operator<< (ostream &s, const tuple<> &t) { return s; }
}


#if defined(LOCAL) || defined(DEBUG)
    #define dump(...) debug_stream << " [ " << __LINE__ << " : " << __FUNCTION__ << " ] " << #__VA_ARGS__ << " : ", dump_func(__VA_ARGS__)
#else
    #define dump(...)
#endif
template <class T> void dump_func(const T &x) { debug_stream << x << '\n'; }
template <class T,class ...Rest> void dump_func(const T &x, Rest ... rest) { debug_stream << x << ","; dump_func(rest...); }
template <class T> void write(const T &x) { std::cout << x << '\n'; }
template <class T, class ...Rest> void write(const T &x, Rest ... rest) { std::cout << x << ' '; write(rest...); }
void writeln() {}
template <class T, class ...Rest> void writeln(const T &x, Rest ... rest) { std::cout << x << '\n'; writeln(rest...); }
#define esc(...) writeln(__VA_ARGS__), exit(0)
template <class P> void read_range(P __first, P __second) { for(P i = __first; i != __second; ++i) std::cin >> *i; }

template <class T> bool chmin(T &x, const T &y) { return x > y ? x = y, true : false; }
template <class T> bool chmax(T &x, const T &y) { return x < y ? x = y, true : false; }
template <class T> constexpr T minf(const T &x, const T &y) { return std::min(x,y); }
template <class T> constexpr T maxf(const T &x, const T &y) { return std::max(x,y); }
constexpr bool odd(int_fast64_t n) { return n & 1; }
constexpr bool even(int_fast64_t n) { return (int)odd(n) ^ 1; }
constexpr bool bit(int_fast64_t n, int e) { return (n >> e) & 1; }
constexpr int_fast64_t mask(int_fast64_t n, int e) { return n & ((1 << e) - 1); }
constexpr int_fast64_t ilog(int_fast64_t x, int_fast64_t b = 2) { return x ? 1 + ilog(x / b, b) : -1; }
constexpr int_fast64_t gcd(int_fast64_t x, int_fast64_t y)
{
    x = x > 0 ? x : -x, y = y > 0 ? y : -y;
    while(y) x ^= y ^= (x %= y) ^= y;
    return x;
}
constexpr int_fast64_t lcm(int_fast64_t x, int_fast64_t y) { return x ? x / gcd(x, y) * y : 0; }
int_fast64_t binry(int_fast64_t ok, int_fast64_t ng, const std::function<bool(int_fast64_t)> &fn)
{
    while (std::abs(ok - ng) > 1)
    {
        int_fast64_t mid = (ok + ng) / 2;
        (fn(mid) ? ok : ng) = mid;
    }
    return ok;
}
template <class A, size_t N, class T> void init(A (&array)[N], const T &val) { std::fill((T*)array, (T*)(array + N), val); }
template <class A, size_t N> void init(A (&array)[N]) { memset(array, 0, sizeof(array)); }
template <class T> std::vector<int> cmprs(const std::vector<T> &v)
{
    std::vector<T> tmp = v; std::vector<int> ret;
    std::sort(tmp.begin(), tmp.end());
    tmp.erase(std::unique(tmp.begin(), tmp.end()), tmp.end());
    for(const T &i : v) ret.emplace_back(std::lower_bound(tmp.begin(), tmp.end() ,i) - tmp.begin());
    return ret;
}
template <class T> std::vector<int> cmprs(const T *__first, const T *__last) { return cmprs(std::vector<T>(__first, __last)); }
void for_subset(int_fast64_t s, const std::function<void(int_fast64_t)> &fn) { int_fast64_t t = s; do { fn(t); } while((--t &= s) != s); }


/* The main code follows. */


signed main()
{
    void solve();
    void input();
    void init();

    int t = 1;

#ifdef LOCAL
    t = 1;
#endif

    // cin >> t;

    while(t--)
    {
        init();
        input();
        solve();
    }
}

template <class Monoid, class act_t>
class Segtree
{
    std::vector<Monoid> data;

public:
    const std::size_t n, N;

    using opr_t = std::function<Monoid(const Monoid &, const Monoid &)>;
    using update_opr_t = std::function<void(Monoid &, const act_t &)>;
    const opr_t opr;
    const update_opr_t update_opr;
    const Monoid identity;

    Segtree(std::size_t _n, const Monoid &_identity, const opr_t &_opr, const update_opr_t &_update_opr)
        : n(_n), N(_n > 1 ? 1 << (32 - __builtin_clz(_n)) : 1), opr(_opr), update_opr(_update_opr), identity(_identity)
    {
        data.assign(N << 1, identity);
    }

    Monoid operator[](std::size_t i) { return data[i + N]; }

    template <class P>
    void copy(P s, P t)
    {
        for (std::size_t i = N; s != t; ++s, ++i)
            data[i] = *s;
        for (std::size_t i = N - 1; i; --i)
            data[i] = opr(data[left(i)], data[right(i)]);
    }

    template <class A>
    void copy(const A &v) { copy(begin(v), end(v)); }

    void init(const Monoid &x)
    {
        for (std::size_t i = 0; i < N; ++i)
            data[i + N] = x;
        for (std::size_t i = N - 1; i; --i)
            data[i] = opr(data[left(i)], data[right(i)]);
    }

    void update(std::size_t idx, const act_t &actor)
    {
        update_opr(data[idx += N], actor);
        while (idx >>= 1)
            data[idx] = opr(data[idx * 2], data[idx * 2 + 1]);
    }

    // operation result of range [a, b).
    Monoid query(std::size_t a, std::size_t b) const
    {
        Monoid lft = identity, rgt = identity;
        a += N, b += N;
        while (a < b)
        {
            if (a & 1)
                lft = opr(lft, data[a++]);
            if (b & 1)
                rgt = opr(data[--b], rgt);
            a >>= 1, b >>= 1;
        }
        return opr(lft, rgt);
    }

    // maximum r where range [idx, r) meets the condition.
    std::size_t right_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f)
    {
        assert(idx < n);
        std::size_t ret = idx;
        Monoid now = identity;
        right_bound(idx, f, 1, 0, N, now, ret);
        return std::min(ret, n);
    }

    // minimum l where range [l, idx) meets the condition.
    std::size_t left_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f)
    {
        assert(idx <= n);
        std::size_t ret = idx;
        Monoid now = identity;
        left_bound(idx, f, 1, 0, N, now, ret);
        return ret;
    }

private:
    constexpr std::size_t left(const std::size_t k) { return k * 2; }

    constexpr std::size_t right(const std::size_t k) { return left(k) ^ 1; }

    constexpr std::size_t parent(const std::size_t k) { return k >> 1; }

    constexpr std::size_t sibling(const std::size_t k) { return k ^ 1; }

    void right_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f, std::size_t k, std::size_t l, std::size_t r, Monoid &now, std::size_t &pos)
    {
        if (idx >= r || l > pos)
            return;
        const std::size_t mid = (l + r) / 2;
        if (l >= idx)
        {
            Monoid nxt = opr(now, data[k]);
            if (f(nxt))
            {
                pos = r;
                now = nxt;
                return;
            }
        }
        if (r - l > 1)
        {
            right_bound(idx, f, left(k), l, mid, now, pos);
            right_bound(idx, f, right(k), mid, r, now, pos);
        }
    }

    void left_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f, std::size_t k, std::size_t l, std::size_t r, Monoid &now, std::size_t &pos)
    {
        if (idx <= l || r < pos)
            return;
        const std::size_t mid = (l + r) / 2;
        if (r <= idx)
        {
            Monoid nxt = opr(data[k], now);
            if (f(nxt))
            {
                pos = l;
                now = nxt;
                return;
            }
        }
        if (r - l > 1)
        {
            left_bound(idx, f, right(k), mid, r, now, pos);
            left_bound(idx, f, left(k), l, mid, now, pos);
        }
    }
};

template <class Monoid, class act_t>
class LazySegtree
{
    std::vector<Monoid> data;
    std::vector<act_t> lazy;
    std::vector<bool> lazyflag;

public:
    const std::size_t n, N;

    using opr_t = std::function<Monoid(const Monoid &, const Monoid &)>;
    using lazy_opr_t = std::function<void(act_t &, const act_t &, std::size_t)>;
    using update_opr_t = std::function<void(Monoid &, const act_t &, std::size_t)>;
    const opr_t opr;
    const lazy_opr_t lazy_opr;
    const update_opr_t update_opr;
    const Monoid identity, lazy_identity;

    constexpr std::size_t adjust_size(const std::size_t n)
    {
        std::size_t d = 0;
        for (std::size_t i = 0; i < 30; ++i)
            if (n >> i & 1)
                d = i;
        return 1 << (d + 1);
    }

    constexpr std::size_t left(const std::size_t k) { return k * 2; }

    constexpr std::size_t right(const std::size_t k) { return left(k) ^ 1; }

    constexpr std::size_t parent(const std::size_t k) { return k >> 1; }

    constexpr std::size_t sibling(const std::size_t k) { return k ^ 1; }

    LazySegtree(std::size_t _n, const Monoid &_identity, const Monoid &_lazy_identity, const opr_t &_opr, const lazy_opr_t &_lazy_opr, const update_opr_t &_update_opr)
        : n(_n), N(_n > 1 ? 1 << (32 - __builtin_clz(_n)) : 1), opr(_opr), lazy_opr(_lazy_opr), update_opr(_update_opr), identity(_identity), lazy_identity(_lazy_identity)
    {
        data.assign(N << 1, identity);
        lazy.assign(N << 1, lazy_identity);
        lazyflag.assign(N << 1, false);
    }

    Monoid operator[](std::size_t i) { return query(i, i + 1); }

    template <class P>
    void copy(P s, P t)
    {
        for (std::size_t i = N; s != t; ++s, ++i)
            data[i] = *s;
        for (std::size_t i = N - 1; i; --i)
            data[i] = opr(data[left(i)], data[right(i)]);
    }

    template <class A>
    void copy(A &v) { copy(begin(v), end(v)); }

    void init(const Monoid &x)
    {
        for (std::size_t i = 0; i < N; ++i)
            data[i + N] = x;
        for (std::size_t i = N - 1; i; --i)
            data[i] = opr(data[left(i)], data[right(i)]);
    }

    void eval(std::size_t k, std::size_t l, std::size_t r)
    {
        if (!lazyflag[k])
            return;
        update_opr(data[k], lazy[k], r - l);
        if (r - l > 1)
        {
            lazy_opr(lazy[left(k)], lazy[k], (r - l) / 2);
            lazy_opr(lazy[right(k)], lazy[k], (r - l) / 2);
            lazyflag[left(k)] = lazyflag[right(k)] = true;
        }
        lazy[k] = lazy_identity;
        lazyflag[k] = false;
    }

    void update(std::size_t a, const act_t &actor) { update(a, a + 1, actor); }

    void update(std::size_t a, std::size_t b, const act_t &actor) { update(a, b, actor, 1, 0, N); }

    void update(std::size_t a, std::size_t b, const act_t &actor, std::size_t k, std::size_t l, std::size_t r)
    {
        eval(k, l, r);
        if (b <= l || r <= a)
            return;
        if (a <= l && r <= b)
        {
            lazy_opr(lazy[k], actor, r - l);
            lazyflag[k] = true;
            eval(k, l, r);
        }
        else
        {
            update(a, b, actor, left(k), l, (l + r) / 2);
            update(a, b, actor, right(k), (l + r) / 2, r);
            data[k] = opr(data[left(k)], data[right(k)]);
        }
    }

    Monoid query(std::size_t a) { return query(a, a + 1); }

    Monoid query(std::size_t a, std::size_t b) { return query(a, b, 1, 0, N); }

    Monoid query(std::size_t a, std::size_t b, std::size_t k, std::size_t l, std::size_t r)
    {
        if (b <= l || r <= a)
            return identity;
        eval(k, l, r);
        if (a <= l && r <= b)
            return data[k];
        return opr(query(a, b, left(k), l, (l + r) / 2), query(a, b, right(k), (l + r) / 2, r));
    }

    std::size_t right_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f)
    {
        assert(idx < n);
        std::size_t ret = idx;
        Monoid now = identity;
        right_bound(idx, f, 1, 0, N, now, ret);
        return std::min(ret, n);
    }

    void right_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f, std::size_t k, std::size_t l, std::size_t r, Monoid &now, std::size_t &pos)
    {
        if (idx >= r || l > pos)
            return;
        eval(k, l, r);
        const std::size_t mid = (l + r) / 2;
        if (l >= idx)
        {
            Monoid nxt = opr(now, data[k]);
            if (f(nxt))
            {
                pos = r;
                now = nxt;
                return;
            }
        }
        if (r - l > 1)
        {
            right_bound(idx, f, left(k), l, mid, now, pos);
            right_bound(idx, f, right(k), mid, r, now, pos);
        }
    }

    std::size_t left_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f)
    {
        assert(idx <= n);
        std::size_t ret = idx;
        Monoid now = identity;
        left_bound(idx, f, 1, 0, N, now, ret);
        return ret;
    }

    void left_bound(std::size_t idx, const std::function<bool(const Monoid &)> &f, std::size_t k, std::size_t l, std::size_t r, Monoid &now, std::size_t &pos)
    {
        if (idx <= l || r < pos)
            return;
        eval(k, l, r);
        const std::size_t mid = (l + r) / 2;
        if (r <= idx)
        {
            Monoid nxt = opr(data[k], now);
            if (f(nxt))
            {
                pos = l;
                now = nxt;
                return;
            }
        }
        if (r - l > 1)
        {
            left_bound(idx, f, right(k), mid, r, now, pos);
            left_bound(idx, f, left(k), l, mid, now, pos);
        }
    }
};

template <class K>
// K must be a field.
struct matrix
{
    std::vector<std::vector<K>> mat;

    matrix() {}
    matrix(std::size_t n) { assign(n, n); }
    matrix(std::size_t h, std::size_t w) { assign(h, w); }
    matrix(const matrix &x) : mat(x.mat) {}

    void resize(std::size_t h, std::size_t w, const K v = K()) { mat.resize(h, std::vector<K>(w, v)); }

    void assign(std::size_t h, std::size_t w, const K v = K()) { mat.assign(h, std::vector<K>(w, v)); }

    std::size_t height() const { return mat.size(); }

    std::size_t width() const { return mat.empty() ? 0 : mat[0].size(); }

    bool is_square() const { return height() == width(); }

    std::vector<K> &operator[](const std::size_t i) { return mat[i]; }

    static matrix identity(std::size_t n)
    {
        matrix ret(n, n);
        for (std::size_t i = 0; i < n; ++i)
            ret[i][i] = K(1);
        return ret;
    }

    matrix operator-() const
    {
        std::size_t h = height(), w = width();
        matrix res(*this);
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                res[i][j] = -mat[i][j];
            }
        }
        return res;
    }

    matrix operator&(const matrix &x) const { return matrix(*this) &= x; }

    matrix operator|(const matrix &x) const { return matrix(*this) |= x; }

    matrix operator^(const matrix &x) const { return matrix(*this) ^= x; }

    matrix operator+(const matrix &x) const { return matrix(*this) += x; }

    matrix operator-(const matrix &x) const { return matrix(*this) -= x; }

    matrix operator*(const matrix &x) const { return matrix(*this) *= x; }

    matrix &operator&=(const matrix &x)
    {
        std::size_t h = height(), w = width();
        assert(h == x.height() and w == x.width());
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                mat[i][j] &= x.mat[i][j];
            }
        }
        return *this;
    }

    matrix &operator|=(const matrix &x)
    {
        std::size_t h = height(), w = width();
        assert(h == x.height() and w == x.width());
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                mat[i][j] |= x.mat[i][j];
            }
        }
        return *this;
    }

    matrix &operator^=(const matrix &x)
    {
        std::size_t h = height(), w = width();
        assert(h == x.height() and w == x.width());
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                mat[i][j] ^= x.mat[i][j];
            }
        }
        return *this;
    }

    matrix &operator+=(const matrix &x)
    {
        std::size_t h = height(), w = width();
        assert(h == x.height() and w == x.width());
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                mat[i][j] += x.mat[i][j];
            }
        }
        return *this;
    }

    matrix &operator-=(const matrix &x)
    {
        std::size_t h = height(), w = width();
        assert(h == x.height() and w == x.width());
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                mat[i][j] -= x.mat[i][j];
            }
        }
        return *this;
    }

    matrix &operator*=(const matrix &x)
    {
        std::size_t l = height(), m = width(), n = x.width();
        assert(m == x.height());
        matrix res(l, n);
        for (std::size_t i = 0; i < l; ++i)
        {
            for (std::size_t j = 0; j < m; ++j)
            {
                for (std::size_t k = 0; k < n; ++k)
                {
                    res[i][k] += mat[i][j] * x.mat[j][k];
                }
            }
        }
        return *this = res;
    }

    friend matrix pow(matrix x, int_fast64_t n)
    {
        assert(x.is_square());
        matrix res = identity(x.height());
        while (n)
        {
            if (n & 1)
                res *= x;
            x *= x;
            n >>= 1;
        }
        return res;
    }

    friend matrix inverse(const matrix &x)
    {
        assert(x.is_square());
        std::size_t n = x.height();
        matrix<K> ext_x(x), idn(identity(n)), ret;
        for (std::size_t i = 0; i < n; ++i)
            ext_x[i].insert(end(ext_x[i]), begin(idn[i]), end(idn[i]));
        std::vector<std::size_t> piv = ext_x.row_canonicalize();
        if (piv.size() < n)
            return matrix<K>();
        ret.mat.resize(n);
        for (std::size_t i = 0; i < n; ++i)
        {
            ret[i] = std::vector<K>(begin(ext_x[i]) + n, end(ext_x[i]));
        }
        return ret;
    }

    std::vector<std::size_t> row_canonicalize()
    {
        std::vector<std::size_t> pivots;
        std::size_t h = height(), w = width(), rank = 0;
        for (std::size_t j = 0; j < w; ++j)
        {
            bool piv = false;
            for (std::size_t i = rank; i < h; ++i)
            {
                if (mat[i][j])
                {
                    if (piv)
                    {
                        K r = -mat[i][j];
                        for (std::size_t k = j; k < w; ++k)
                        {
                            mat[i][k] += mat[rank][k] * r;
                        }
                    }
                    else
                    {
                        swap(mat[rank], mat[i]);
                        K r = mat[rank][j];
                        for (std::size_t k = j; k < w; ++k)
                        {
                            mat[rank][k] /= r;
                        }
                        for (std::size_t k = 0; k < rank; ++k)
                        {
                            r = -mat[k][j];
                            for (std::size_t l = j; l < w; ++l)
                            {
                                mat[k][l] += mat[rank][l] * r;
                            }
                        }
                        piv = true;
                    }
                }
            }
            if (piv)
            {
                pivots.emplace_back(j);
                ++rank;
            }
        }
        return pivots;
    }

    K det() const
    {
        matrix<K> x(*this);
        assert(is_square());
        std::size_t n = height();
        K res(1);
        for (std::size_t j = 0; j < n; ++j)
        {
            bool piv = false;
            for (std::size_t i = j; i < n; ++i)
            {
                if (x[i][j])
                {
                    if (piv)
                    {
                        const K r = -x[i][j];
                        for (std::size_t k = j; k < n; ++k)
                        {
                            x[i][k] += x[j][k] * r;
                        }
                    }
                    else
                    {
                        swap(x[i], x[j]);
                        if (i != j)
                            res = -res;
                        const K r = x[j][j];
                        res *= r;
                        for (std::size_t k = j; k < n; ++k)
                        {
                            x[j][k] /= r;
                        }
                        piv = true;
                    }
                }
            }
            if (not piv)
            {
                return K(0);
            }
        }
        return res;
    }

    friend std::istream &operator>>(std::istream &s, matrix &x)
    {
        std::size_t h = x.height(), w = x.width();
        for (std::size_t i = 0; i < h; ++i)
        {
            for (std::size_t j = 0; j < w; ++j)
            {
                s >> x[i][j];
            }
        }
        return s;
    }

    friend std::ostream &operator<<(std::ostream &s, const matrix &x)
    {
        std::size_t h = x.height(), w = x.width();
        for (std::size_t i = 0; i < h; ++i)
        {
            if (i)
                s << "\n";
            for (std::size_t j = 0; j < w; ++j)
            {
                s << (j ? " " : "") << x.mat[i][j];
            }
        }
        return s;
    }
};

int n;
int qry;
using mat_t = matrix<i64>;
LazySegtree<mat_t,mat_t> seg(1u<<17,mat_t(3,1),mat_t::identity(3),plus<mat_t>(),[](mat_t &x, const mat_t y,int w){x=y*x;},[](mat_t &x,const mat_t y,int w){x=y*x;});


void init()
{}


void input()
{
    cin >> n >> qry;
    vector<mat_t> ini(n);
    for(int i=0; i<n; i++)
    {
        int a; std::cin >> a;
        mat_t pb(3,1);
        pb[0][0]=a;
        pb[(a&1)+1][0]=1;
        ini[i]=pb;
    }
    seg.copy(ini);
}

void solve()
{
    while(qry--)
    {
        int typ,l,r;
        std::cin >> typ >> l >> r;
        l--;
        if(typ==1)
        {
            mat_t up(mat_t::identity(3));
            up[0]={0,0,1};
            dump(up);
            seg.update(l,r,up);
        }
        else if(typ==2)
        {
            mat_t up(mat_t::identity(3));
            int x; std::cin >> x;
            up[0]={1,x,x};
            if(x&1)
            {
                swap(up[1],up[2]);
            }
            dump(up);
            seg.update(l,r,up);
        }
        else
        {
            auto res=seg.query(l,r);
            dump(res);
            std::cout << res[0][0] << "\n";
        }
    }
}