#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>


#include <iostream>
#include <string>
#include <utility>
#include <vector>

class JsonDumper {
    struct KeyValue {
        std::string key;
        std::string value;
    };

    std::vector<KeyValue> _items;

    bool dump_at_end = false;

public:
    JsonDumper(bool dump_at_end_ = false) : dump_at_end(dump_at_end_) {}

    ~JsonDumper() {
        if (dump_at_end) std::cout << dump() << std::endl;
    }

    void set_dump_at_end() { dump_at_end = true; }

    void operator()(const std::string &key, const std::string &value) {
        _items.push_back(KeyValue{key, "\"" + value + "\""});
    }

    template <class T> void operator()(const std::string &key, T value) {
        _items.push_back(KeyValue{key, std::to_string(value)});
    }

    std::string dump() const {
        std::string ret = "{\n";

        if (!_items.empty()) {
            for (const auto &[k, v] : _items) ret += "    \"" + k + "\": " + v + ",\n";

            ret.erase(ret.end() - 2);
        }

        ret += "}";
        return ret;
    }
} jdump;

#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for (int i = (begin), i##_end_ = (end); i < i##_end_; i++)
#define IFOR(i, begin, end) for (int i = (end)-1, i##_begin_ = (begin); i >= i##_begin_; i--)
#define REP(i, n) FOR(i, 0, n)
#define IREP(i, n) IFOR(i, 0, n)

template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }

int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <class T1, class T2> T1 floor_div(T1 num, T2 den) {
    return (num > 0 ? num / den : -((-num + den - 1) / den));
}
template <class T1, class T2> std::pair<T1, T2> operator+(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) {
    return std::make_pair(l.first + r.first, l.second + r.second);
}
template <class T1, class T2> std::pair<T1, T2> operator-(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) {
    return std::make_pair(l.first - r.first, l.second - r.second);
}
template <class T> std::vector<T> sort_unique(std::vector<T> vec) {
    sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end());
    return vec;
}
template <class T> int arglb(const std::vector<T> &v, const T &x) {
    return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x));
}
template <class T> int argub(const std::vector<T> &v, const T &x) {
    return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x));
}
template <class IStream, class T> IStream &operator>>(IStream &is, std::vector<T> &vec) {
    for (auto &v : vec) is >> v;
    return is;
}

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec);
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr);
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp);
template <class OStream, class TK, class TV, class TH>
OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp);
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl);

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec) {
    os << '[';
    for (auto v : vec) os << v << ',';
    os << ']';
    return os;
}
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr) {
    os << '[';
    for (auto v : arr) os << v << ',';
    os << ']';
    return os;
}
template <class... T> std::istream &operator>>(std::istream &is, std::tuple<T...> &tpl) {
    std::apply([&is](auto &&...args) { ((is >> args), ...); }, tpl);
    return is;
}
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl) {
    os << '(';
    std::apply([&os](auto &&...args) { ((os << args << ','), ...); }, tpl);
    return os << ')';
}
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec) {
    os << '{';
    for (auto v : vec) os << v << ',';
    os << '}';
    return os;
}
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec) {
    os << "deq[";
    for (auto v : vec) os << v << ',';
    os << ']';
    return os;
}
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec) {
    os << '{';
    for (auto v : vec) os << v << ',';
    os << '}';
    return os;
}
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec) {
    os << '{';
    for (auto v : vec) os << v << ',';
    os << '}';
    return os;
}
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec) {
    os << '{';
    for (auto v : vec) os << v << ',';
    os << '}';
    return os;
}
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa) {
    return os << '(' << pa.first << ',' << pa.second << ')';
}
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp) {
    os << '{';
    for (auto v : mp) os << v.first << "=>" << v.second << ',';
    os << '}';
    return os;
}
template <class OStream, class TK, class TV, class TH>
OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp) {
    os << '{';
    for (auto v : mp) os << v.first << "=>" << v.second << ',';
    os << '}';
    return os;
}

#ifdef HITONANODE_LOCAL
const std::string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m",
                  BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m",
                  NORMAL_FAINT = "\033[0;2m";
#define dbg(x)                                                                                                        \
    std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") "        \
              << __FILE__ << COLOR_RESET << std::endl
#define dbgif(cond, x)                                                                                                \
    ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__      \
                        << ") " << __FILE__ << COLOR_RESET << std::endl                                               \
            : std::cerr)
#else
#define dbg(x) 0
#define dbgif(cond, x) 0
#endif

#ifdef BENCHMARK
#define dump_onlinejudge(x) 0
struct setenv {
    setenv() { jdump.set_dump_at_end(); }
} setenv_;
#else
#define dump_onlinejudge(x) (std::cout << (x) << std::endl)
#endif

#include <vector>


#include <cstdint>

uint32_t rand_int() // XorShift random integer generator
{
    static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123;
    uint32_t t = x ^ (x << 11);
    x = y;
    y = z;
    z = w;
    return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8));
}
double rand_double() { return (double)rand_int() / UINT32_MAX; }


#include <algorithm>
#include <cassert>
#include <deque>
#include <fstream>
#include <functional>
#include <limits>
#include <queue>
#include <string>
#include <tuple>
#include <utility>
#include <vector>

template <typename T, T INF = std::numeric_limits<T>::max() / 2, int INVALID = -1>
struct shortest_path {
    int V, E;
    bool single_positive_weight;
    T wmin, wmax;

    std::vector<std::pair<int, T>> tos;
    std::vector<int> head;
    std::vector<std::tuple<int, int, T>> edges;

    void build_() {
        if (int(tos.size()) == E and int(head.size()) == V + 1) return;
        tos.resize(E);
        head.assign(V + 1, 0);
        for (const auto &e : edges) ++head[std::get<0>(e) + 1];
        for (int i = 0; i < V; ++i) head[i + 1] += head[i];
        auto cur = head;
        for (const auto &e : edges) {
            tos[cur[std::get<0>(e)]++] = std::make_pair(std::get<1>(e), std::get<2>(e));
        }
    }

    shortest_path(int V = 0) : V(V), E(0), single_positive_weight(true), wmin(0), wmax(0) {}
    void add_edge(int s, int t, T w) {
        assert(0 <= s and s < V);
        assert(0 <= t and t < V);
        edges.emplace_back(s, t, w);
        ++E;
        if (w > 0 and wmax > 0 and wmax != w) single_positive_weight = false;
        wmin = std::min(wmin, w);
        wmax = std::max(wmax, w);
    }

    void add_bi_edge(int u, int v, T w) {
        add_edge(u, v, w);
        add_edge(v, u, w);
    }

    std::vector<T> dist;
    std::vector<int> prev;

    using Pque = std::priority_queue<std::pair<T, int>, std::vector<std::pair<T, int>>,
                                     std::greater<std::pair<T, int>>>;
    template <class Heap = Pque> void dijkstra(int s, int t = INVALID) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF);
        prev.assign(V, INVALID);
        dist[s] = 0;
        Heap pq;
        pq.emplace(0, s);
        while (!pq.empty()) {
            T d;
            int v;
            std::tie(d, v) = pq.top();
            pq.pop();
            if (t == v) return;
            if (dist[v] < d) continue;
            for (int e = head[v]; e < head[v + 1]; ++e) {
                const auto &nx = tos[e];
                T dnx = d + nx.second;
                if (dist[nx.first] > dnx) {
                    dist[nx.first] = dnx, prev[nx.first] = v;
                    pq.emplace(dnx, nx.first);
                }
            }
        }
    }

    void dijkstra_vquad(int s, int t = INVALID) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF);
        prev.assign(V, INVALID);
        dist[s] = 0;
        std::vector<char> fixed(V, false);
        while (true) {
            int r = INVALID;
            T dr = INF;
            for (int i = 0; i < V; i++) {
                if (!fixed[i] and dist[i] < dr) r = i, dr = dist[i];
            }
            if (r == INVALID or r == t) break;
            fixed[r] = true;
            int nxt;
            T dx;
            for (int e = head[r]; e < head[r + 1]; ++e) {
                std::tie(nxt, dx) = tos[e];
                if (dist[nxt] > dist[r] + dx) dist[nxt] = dist[r] + dx, prev[nxt] = r;
            }
        }
    }

    bool bellman_ford(int s, int nb_loop) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF), prev.assign(V, INVALID);
        dist[s] = 0;
        for (int l = 0; l < nb_loop; l++) {
            bool upd = false;
            for (int v = 0; v < V; v++) {
                if (dist[v] == INF) continue;
                for (int e = head[v]; e < head[v + 1]; ++e) {
                    const auto &nx = tos[e];
                    T dnx = dist[v] + nx.second;
                    if (dist[nx.first] > dnx) dist[nx.first] = dnx, prev[nx.first] = v, upd = true;
                }
            }
            if (!upd) return true;
        }
        return false;
    }

    void spfa(int s) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF);
        prev.assign(V, INVALID);
        dist[s] = 0;
        std::deque<int> q;
        std::vector<char> in_queue(V);
        q.push_back(s), in_queue[s] = 1;
        while (!q.empty()) {
            int now = q.front();
            q.pop_front(), in_queue[now] = 0;
            for (int e = head[now]; e < head[now + 1]; ++e) {
                const auto &nx = tos[e];
                T dnx = dist[now] + nx.second;
                int nxt = nx.first;
                if (dist[nxt] > dnx) {
                    dist[nxt] = dnx;
                    if (!in_queue[nxt]) {
                        if (q.size() and dnx < dist[q.front()]) { // Small label first optimization
                            q.push_front(nxt);
                        } else {
                            q.push_back(nxt);
                        }
                        prev[nxt] = now, in_queue[nxt] = 1;
                    }
                }
            }
        }
    }

    void zero_one_bfs(int s, int t = INVALID) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF), prev.assign(V, INVALID);
        dist[s] = 0;
        std::vector<int> q(V * 4);
        int ql = V * 2, qr = V * 2;
        q[qr++] = s;
        while (ql < qr) {
            int v = q[ql++];
            if (v == t) return;
            for (int e = head[v]; e < head[v + 1]; ++e) {
                const auto &nx = tos[e];
                T dnx = dist[v] + nx.second;
                if (dist[nx.first] > dnx) {
                    dist[nx.first] = dnx, prev[nx.first] = v;
                    if (nx.second) {
                        q[qr++] = nx.first;
                    } else {
                        q[--ql] = nx.first;
                    }
                }
            }
        }
    }

    void dial(int s, int t = INVALID) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF), prev.assign(V, INVALID);
        dist[s] = 0;
        std::vector<std::vector<std::pair<int, T>>> q(wmax + 1);
        q[0].emplace_back(s, dist[s]);
        int ninq = 1;

        int cur = 0;
        T dcur = 0;
        for (; ninq; ++cur, ++dcur) {
            if (cur == wmax + 1) cur = 0;
            while (!q[cur].empty()) {
                int v = q[cur].back().first;
                T dnow = q[cur].back().second;
                q[cur].pop_back(), --ninq;
                if (v == t) return;
                if (dist[v] < dnow) continue;

                for (int e = head[v]; e < head[v + 1]; ++e) {
                    const auto &nx = tos[e];
                    T dnx = dist[v] + nx.second;
                    if (dist[nx.first] > dnx) {
                        dist[nx.first] = dnx, prev[nx.first] = v;
                        int nxtcur = cur + int(nx.second);
                        if (nxtcur >= int(q.size())) nxtcur -= q.size();
                        q[nxtcur].emplace_back(nx.first, dnx), ++ninq;
                    }
                }
            }
        }
    }

    bool dag_solver(int s) {
        assert(0 <= s and s < V);
        build_();
        dist.assign(V, INF), prev.assign(V, INVALID);
        dist[s] = 0;
        std::vector<int> indeg(V, 0);
        std::vector<int> q(V * 2);
        int ql = 0, qr = 0;
        q[qr++] = s;
        while (ql < qr) {
            int now = q[ql++];
            for (int e = head[now]; e < head[now + 1]; ++e) {
                const auto &nx = tos[e];
                ++indeg[nx.first];
                if (indeg[nx.first] == 1) q[qr++] = nx.first;
            }
        }
        ql = qr = 0;
        q[qr++] = s;
        while (ql < qr) {
            int now = q[ql++];
            for (int e = head[now]; e < head[now + 1]; ++e) {
                const auto &nx = tos[e];
                --indeg[nx.first];
                if (dist[nx.first] > dist[now] + nx.second)
                    dist[nx.first] = dist[now] + nx.second, prev[nx.first] = now;
                if (indeg[nx.first] == 0) q[qr++] = nx.first;
            }
        }
        return *max_element(indeg.begin(), indeg.end()) == 0;
    }

    std::vector<int> retrieve_path(int goal) const {
        assert(int(prev.size()) == V);
        assert(0 <= goal and goal < V);
        if (dist[goal] == INF) return {};
        std::vector<int> ret{goal};
        while (prev[goal] != INVALID) {
            goal = prev[goal];
            ret.push_back(goal);
        }
        std::reverse(ret.begin(), ret.end());
        return ret;
    }

    void solve(int s, int t = INVALID) {
        if (wmin >= 0) {
            if (single_positive_weight) {
                zero_one_bfs(s, t);
            } else if (wmax <= 10) {
                dial(s, t);
            } else {
                if ((long long)V * V < (E << 4)) {
                    dijkstra_vquad(s, t);
                } else {
                    dijkstra(s, t);
                }
            }
        } else {
            bellman_ford(s, V);
        }
    }

    std::vector<std::vector<T>> floyd_warshall() {
        build_();
        std::vector<std::vector<T>> dist2d(V, std::vector<T>(V, INF));
        for (int i = 0; i < V; i++) {
            dist2d[i][i] = 0;
            for (const auto &e : edges) {
                int s = std::get<0>(e), t = std::get<1>(e);
                dist2d[s][t] = std::min(dist2d[s][t], std::get<2>(e));
            }
        }
        for (int k = 0; k < V; k++) {
            for (int i = 0; i < V; i++) {
                if (dist2d[i][k] == INF) continue;
                for (int j = 0; j < V; j++) {
                    if (dist2d[k][j] == INF) continue;
                    dist2d[i][j] = std::min(dist2d[i][j], dist2d[i][k] + dist2d[k][j]);
                }
            }
        }
        return dist2d;
    }

    void to_dot(std::string filename = "shortest_path") const {
        std::ofstream ss(filename + ".DOT");
        ss << "digraph{\n";
        build_();
        for (int i = 0; i < V; i++) {
            for (int e = head[i]; e < head[i + 1]; ++e) {
                ss << i << "->" << tos[e].first << "[label=" << tos[e].second << "];\n";
            }
        }
        ss << "}\n";
        ss.close();
        return;
    }
};

#include <algorithm>
#include <utility>
#include <vector>


#include <vector>



#include <algorithm>
#include <limits>
#include <utility>
#include <vector>

template <class DistanceMatrix>
std::vector<std::pair<int, int>> mst_edges(const DistanceMatrix &dist) {
    using T = decltype((*dist.adjacents(0).begin()).second);

    if (dist.n() <= 1) return {};

    std::vector<T> dp(dist.n(), std::numeric_limits<T>::max());
    std::vector<int> prv(dist.n(), -1);
    std::vector<int> used(dist.n());
    std::vector<std::pair<int, int>> ret(dist.n() - 1);

    for (int t = 0; t < dist.n(); ++t) {
        int x = std::min_element(dp.cbegin(), dp.cend()) - dp.cbegin();
        dp.at(x) = std::numeric_limits<T>::max();
        used.at(x) = 1;
        if (t > 0) ret.at(t - 1) = {prv.at(x), x};

        for (auto [y, len] : dist.adjacents(x)) {
            if (!used.at(y) and len < dp.at(y)) dp.at(y) = len, prv.at(y) = x;
        }
    }

    return ret;
}

template <class DistanceMatrix> auto calc_lkh_alpha(const DistanceMatrix &dist) {
    using T = decltype((*dist.adjacents(0).begin()).second);

    std::vector<std::vector<int>> to(dist.n());

    for (auto [s, t] : mst_edges(dist)) {
        to.at(s).push_back(t);
        to.at(t).push_back(s);
    }

    std::vector ret(dist.n(), std::vector<T>(dist.n()));

    for (int s = 0; s < dist.n(); ++s) {
        auto rec = [&](auto &&self, int now, int prv, T hi) -> void {
            ret.at(s).at(now) = dist(s, now) - hi;
            for (int nxt : to.at(now)) {
                if (nxt == prv) continue;
                self(self, nxt, now, std::max(hi, dist(now, nxt)));
            }
        };
        rec(rec, s, -1, T());
    }

    int best_one = -1;
    T longest_2nd_nearest = T();

    for (int one = 0; one < dist.n(); ++one) {
        if (to.at(one).size() != 1) continue;
        const int ng = to.at(one).front();
        bool found = false;
        T second_nearest = T();

        for (auto [v, len] : dist.adjacents(one)) {
            if (v == ng) continue;
            if (!found) {
                found = true, second_nearest = len;
            } else if (len < second_nearest) {
                second_nearest = len;
            }
        }

        if (found and (best_one < 0 or second_nearest > longest_2nd_nearest))
            best_one = one, longest_2nd_nearest = second_nearest;
    }

    if (best_one != -1) {
        for (auto [v, len] : dist.adjacents(best_one)) {
            if (v == to.at(best_one).front()) continue;
            ret.at(best_one).at(v) = ret.at(v).at(best_one) = len - longest_2nd_nearest;
        }
    }

    return ret;
}

template <class T> class csr_distance_matrix {

    int _rows = 0;
    std::vector<int> begins;
    std::vector<std::pair<int, T>> vals;

public:
    csr_distance_matrix() : csr_distance_matrix({}, 0) {}

    csr_distance_matrix(const std::vector<std::tuple<int, int, T>> &init, int rows)
        : _rows(rows), begins(rows + 1) {
        std::vector<int> degs(rows);
        for (const auto &p : init) ++degs.at(std::get<0>(p));

        for (int i = 0; i < rows; ++i) begins.at(i + 1) = begins.at(i) + degs.at(i);

        vals.resize(init.size(), std::make_pair(-1, T()));
        for (auto [i, j, w] : init) vals.at(begins.at(i + 1) - (degs.at(i)--)) = {j, w};
    }

    void apply_pi(const std::vector<T> &pi) {
        for (int i = 0; i < n(); ++i) {
            for (auto &[j, d] : adjacents(i)) d += pi.at(i) + pi.at(j);
        }
    }

    int n() const noexcept { return _rows; }

    struct adjacents_sequence {
        csr_distance_matrix *ptr;
        int from;

        using iterator = typename std::vector<std::pair<int, T>>::iterator;
        iterator begin() { return std::next(ptr->vals.begin(), ptr->begins.at(from)); }
        iterator end() { return std::next(ptr->vals.begin(), ptr->begins.at(from + 1)); }
    };

    struct const_adjacents_sequence {
        const csr_distance_matrix *ptr;
        const int from;

        using const_iterator = typename std::vector<std::pair<int, T>>::const_iterator;
        const_iterator begin() const {
            return std::next(ptr->vals.cbegin(), ptr->begins.at(from));
        }
        const_iterator end() const {
            return std::next(ptr->vals.cbegin(), ptr->begins.at(from + 1));
        }
    };

    adjacents_sequence adjacents(int from) { return {this, from}; }

    const_adjacents_sequence adjacents(int from) const { return {this, from}; }
    const_adjacents_sequence operator()(int from) const { return {this, from}; }
};

template <class DistanceMatrix> auto build_adjacent_info(const DistanceMatrix &dist, int sz) {
    using T = decltype((*dist.adjacents(0).begin()).second);

    const std::vector<std::vector<T>> alpha = calc_lkh_alpha(dist);

    std::vector<std::tuple<int, int, T>> adjacent_edges;

    std::vector<std::tuple<T, T, int>> candidates;
    for (int i = 0; i < dist.n(); ++i) {
        candidates.clear();
        for (auto [j, d] : dist.adjacents(i)) {
            if (i != j) candidates.emplace_back(alpha.at(i).at(j), d, j);
        }

        const int final_sz = std::min<int>(sz, candidates.size());
        std::nth_element(candidates.begin(), candidates.begin() + final_sz, candidates.end());

        candidates.resize(final_sz);
        std::sort(candidates.begin(), candidates.end(),
                  [&](const auto &l, const auto &r) { return std::get<1>(l) < std::get<1>(r); });
        for (auto [alpha, dij, j] : candidates) adjacent_edges.emplace_back(i, j, dij);
    }
    return csr_distance_matrix(adjacent_edges, dist.n());
}

#include <algorithm>
#include <array>
#include <random>
#include <vector>

template <class DistanceMatrix, class Adjacents> struct SymmetricTSP {
    DistanceMatrix dist;
    Adjacents adjacents;
    using T = decltype((*dist.adjacents(0).begin()).second);

    struct Solution {
        T cost;
        std::vector<int> path;

        template <class OStream> friend OStream &operator<<(OStream &os, const Solution &x) {
            os << "[cost=" << x.cost << ", path=(";
            for (int i : x.path) os << i << ",";
            return os << x.path.front() << ")]";
        }
    };

    T eval(const Solution &sol) const {
        T ret = T();
        int now = sol.path.back();
        for (int nxt : sol.path) ret += dist(now, nxt), now = nxt;
        return ret;
    }

    SymmetricTSP(const DistanceMatrix &distance_matrix, const Adjacents &adjacents)
        : dist(distance_matrix), adjacents(adjacents) {}

    Solution nearest_neighbor(int init) const {
        if (n() == 0) return {T(), {}};
        int now = init;
        std::vector<int> ret{now}, alive(n(), 1);
        T len = T();
        ret.reserve(n());
        alive.at(now) = 0;
        while (int(ret.size()) < n()) {
            int nxt = -1;
            for (int i = 0; i < n(); ++i) {
                if (alive.at(i) and (nxt < 0 or dist(now, i) < dist(now, nxt))) nxt = i;
            }
            ret.push_back(nxt);
            alive.at(nxt) = 0;
            len += dist(now, nxt);
            now = nxt;
        }
        len += dist(ret.back(), ret.front());
        return Solution{len, ret};
    }

    void two_opt(Solution &sol) const {
        static std::vector<int> v_to_i;
        v_to_i.resize(n());
        for (int i = 0; i < n(); ++i) v_to_i.at(sol.path.at(i)) = i;
        while (true) {
            bool updated = false;
            for (int i = 0; i < n() and !updated; ++i) {
                const int u = sol.path.at(i), nxtu = sol.path.at(modn(i + 1));
                const T dunxtu = dist(u, nxtu);

                for (auto [v, duv] : adjacents(u)) {
                    if (duv >= dunxtu) break;
                    int j = v_to_i.at(v), nxtv = sol.path.at(modn(j + 1));
                    T diff = duv + dist(nxtu, nxtv) - dunxtu - dist(v, nxtv);
                    if (diff < 0) {
                        sol.cost += diff;
                        int l, r;
                        if (modn(j - i) < modn(i - j)) {
                            l = modn(i + 1), r = j;
                        } else {
                            l = modn(j + 1), r = i;
                        }
                        while (l != r) {
                            std::swap(sol.path.at(l), sol.path.at(r));
                            v_to_i.at(sol.path.at(l)) = l;
                            v_to_i.at(sol.path.at(r)) = r;
                            l = modn(l + 1);
                            if (l == r) break;
                            r = modn(r - 1);
                        }
                        updated = true;
                        break;
                    }
                }
                if (updated) break;

                for (auto [nxtv, dnxtunxtv] : adjacents(nxtu)) {
                    if (dnxtunxtv >= dunxtu) break;
                    int j = modn(v_to_i.at(nxtv) - 1), v = sol.path.at(j);
                    T diff = dist(u, v) + dnxtunxtv - dunxtu - dist(v, nxtv);
                    if (diff < 0) {
                        sol.cost += diff;
                        int l, r;
                        if (modn(j - i) < modn(i - j)) {
                            l = modn(i + 1), r = j;
                        } else {
                            l = modn(j + 1), r = i;
                        }
                        while (l != r) {
                            std::swap(sol.path.at(l), sol.path.at(r));
                            v_to_i.at(sol.path.at(l)) = l;
                            v_to_i.at(sol.path.at(r)) = r;
                            l = modn(l + 1);
                            if (l == r) break;
                            r = modn(r - 1);
                        }
                        updated = true;
                        break;
                    }
                }
            }
            if (!updated) break;
        }
    }

    bool three_opt(Solution &sol) const {
        static std::vector<int> v_to_i;
        v_to_i.resize(n());
        for (int i = 0; i < n(); ++i) v_to_i.at(sol.path.at(i)) = i;

        auto check_uvw_order = [](int u, int v, int w) {
            int i = v_to_i.at(u);
            int j = v_to_i.at(v);
            int k = v_to_i.at(w);
            if (i < j and j < k) return true;
            if (j < k and k < i) return true;
            if (k < i and i < j) return true;
            return false;
        };

        auto rev = [&](const int u, const int v) -> void {
            int l = v_to_i.at(u), r = v_to_i.at(v);
            while (l != r) {
                std::swap(sol.path.at(l), sol.path.at(r));
                l = modn(l + 1);
                if (l == r) break;
                r = modn(r - 1);
            }
        };

        static int i = 0;
        for (int nseen = 0; nseen < n(); ++nseen, i = modn(i + 1)) {
            const int u = sol.path.at(modn(i - 1)), nxtu = sol.path.at(i);
            const T dunxtu = dist(u, nxtu);

            for (const auto &[nxtv, dunxtv] : adjacents(u)) {
                if (dunxtv >= dunxtu) break;
                const int v = sol.path.at(modn(v_to_i.at(nxtv) - 1));
                const T dvnxtv = dist(v, nxtv);

                for (const auto &[nxtw, dvnxtw] : adjacents(v)) {
                    if (nxtw == nxtv or nxtw == nxtu) continue;
                    if (dunxtv + dvnxtw >= dunxtu + dvnxtv) break;
                    const int w = sol.path.at(modn(v_to_i.at(nxtw) - 1));

                    if (!check_uvw_order(u, v, w)) continue;

                    const T current = dunxtu + dvnxtv + dist(w, nxtw);
                    if (T diff = dunxtv + dist(w, nxtu) + dvnxtw - current; diff < T()) {
                        sol.cost += diff;
                        rev(nxtu, v);
                        rev(nxtv, w);
                        rev(nxtw, u);
                        return true;
                    }
                }

                for (const auto &[w, dvw] : adjacents(v)) {
                    if (dunxtv + dvw >= dunxtu + dvnxtv) break;
                    if (!check_uvw_order(u, v, w)) continue;

                    const int nxtw = sol.path.at(modn(v_to_i.at(w) + 1));

                    const T current = dunxtu + dvnxtv + dist(w, nxtw);

                    if (T diff = dunxtv + dvw + dist(nxtu, nxtw) - current; diff < T()) {
                        sol.cost += diff;
                        rev(nxtw, u);
                        rev(nxtv, w);
                        return true;
                    }
                }
            }

            for (const auto &[nxtv, dnxtunxtv] : adjacents(nxtu)) {
                if (dnxtunxtv >= dunxtu) break;
                const int v = sol.path.at(modn(v_to_i.at(nxtv) - 1));
                const T dvnxtv = dist(v, nxtv);

                for (const auto &[nxtw, dvnxtw] : adjacents(v)) {
                    const int w = sol.path.at(modn(v_to_i.at(nxtw) - 1));
                    if (dnxtunxtv + dvnxtw >= dunxtu + dvnxtv) break;
                    if (!check_uvw_order(u, v, w)) continue;

                    const T current = dunxtu + dvnxtv + dist(w, nxtw);
                    if (T diff = dist(u, w) + dnxtunxtv + dvnxtw - current; diff < T()) {
                        sol.cost += diff;
                        rev(nxtu, v);
                        rev(nxtw, u);
                        return true;
                    }
                }
            }

            for (const auto &[v, duv] : adjacents(u)) {
                if (duv >= dunxtu) break;
                const int nxtv = sol.path.at(modn(v_to_i.at(v) + 1));
                const T dvnxtv = dist(v, nxtv);

                for (const auto &[nxtw, dnxtvnxtw] : adjacents(nxtv)) {
                    const int w = sol.path.at(modn(v_to_i.at(nxtw) - 1));
                    if (duv + dnxtvnxtw >= dunxtu + dvnxtv) break;
                    if (!check_uvw_order(u, v, w)) continue;

                    const T current = dunxtu + dvnxtv + dist(w, nxtw);
                    if (T diff = duv + dist(nxtu, w) + dnxtvnxtw - current; diff < T()) {
                        sol.cost += diff;
                        rev(nxtu, v);
                        rev(nxtv, w);
                        return true;
                    }
                }
            }
        }
        return false;
    }

    template <class Rng> bool double_bridge(Solution &sol, Rng &rng) const {
        if (n() < 8) return false;

        std::vector<int> &p = sol.path;
        int rand_rot = std::uniform_int_distribution<int>(0, n() - 1)(rng);
        std::rotate(p.begin(), p.begin() + rand_rot, p.end());

        static std::array<int, 3> arr;
        for (int &y : arr) y = std::uniform_int_distribution<int>(2, n() - 6)(rng);
        std::sort(arr.begin(), arr.end());
        const int i = arr.at(0), j = arr.at(1) + 2, k = arr.at(2) + 4;
        static std::array<T, 2> diffs;
        for (int d = 0; d < 2; ++d) {
            int u = p.at(n() - 1), nxtu = p.at(0);
            int v = p.at(i - 1), nxtv = p.at(i);
            int w = p.at(j - 1), nxtw = p.at(j);
            int x = p.at(k - 1), nxtx = p.at(k);
            diffs.at(d) = dist(u, nxtu) + dist(v, nxtv) + dist(w, nxtw) + dist(x, nxtx);
            if (d == 1) break;
            std::reverse(p.begin(), p.begin() + i);
            std::reverse(p.begin() + i, p.begin() + j);
            std::reverse(p.begin() + j, p.begin() + k);
            std::reverse(p.begin() + k, p.end());
        }
        sol.cost += diffs.at(1) - diffs.at(0);
        return true;
    }

    int n() const noexcept { return dist.n(); }
    int modn(int x) const noexcept {
        if (x < 0) return x + n();
        if (x >= n()) return x - n();
        return x;
    }
};

template <class T> class dense_distance_matrix {
    int _n;
    std::vector<T> _d;

public:
    dense_distance_matrix(const std::vector<std::vector<T>> &distance_vecvec)
        : _n(distance_vecvec.size()) {
        _d.reserve(n() * n());
        for (const auto &vec : distance_vecvec) _d.insert(end(_d), begin(vec), end(vec));
    }

    template <class U> void apply_pi(const std::vector<U> &pi) {
        for (int i = 0; i < n(); ++i) {
            for (int j = 0; j < n(); ++j) _d.at(i * n() + j) += pi.at(i) + pi.at(j);
        }
    }

    int n() const noexcept { return _n; }

    T dist(int s, int t) const { return _d.at(s * n() + t); }
    T operator()(int s, int t) const { return dist(s, t); }

    struct adjacents_sequence {
        const dense_distance_matrix *ptr;
        int from;
        struct iterator {
            const dense_distance_matrix *ptr;
            int from;
            int to;
            iterator operator++() { return {ptr, from, to++}; }
            std::pair<int, T> operator*() const { return {to, ptr->dist(from, to)}; }
            bool operator!=(const iterator &rhs) const {
                return to != rhs.to or ptr != rhs.ptr or from != rhs.from;
            }
        };
        iterator begin() const { return iterator{ptr, from, 0}; }
        iterator end() const { return iterator{ptr, from, ptr->n()}; }
    };

    adjacents_sequence adjacents(int from) const { return {this, from}; }
};


struct Solver {
    static constexpr int N = 100, M = 8;
    static constexpr int alpha = 5;
    static constexpr int UB = 1000;

    std::vector<int> X, Y;
    std::vector<int> C, D;

    std::vector<std::vector<long long>> direct_distances;

    int calc_direct_dist(int i, int j) const {
        int dx = X.at(i) - X.at(j);
        int dy = Y.at(i) - Y.at(j);
        return (dx * dx + dy * dy) * (alpha * alpha);
    }

    int calc_s2s_dist(int p, int q) const {
        int dx = C.at(p) - C.at(q);
        int dy = D.at(p) - D.at(q);
        return (dx * dx + dy * dy);
    }

    int calc_p2s_dist(int i, int p) const {
        int dx = X.at(i) - C.at(p);
        int dy = Y.at(i) - D.at(p);
        return (dx * dx + dy * dy) * alpha;
    }

    using Matrix = dense_distance_matrix<long long>;
    using TSP = SymmetricTSP<Matrix, csr_distance_matrix<long long>>;

    void three_opt(const TSP &tsp, TSP::Solution &sol) {
        do {
            tsp.two_opt(sol);
        } while (tsp.three_opt(sol)); // three_opt() は一度 3-opt による改善に成功した時点で return
    }

    TSP::Solution sol;

    Solver(const std::vector<int> &a, const std::vector<int> &b) : X(a), Y(b), C(M), D(M) {
        direct_distances.assign(N, std::vector<long long>(N));
        for (int i = 0; i < N; ++i) {
            direct_distances.at(i).at(i) = 0;
            for (int j = i + 1; j < N; ++j) {
                direct_distances.at(i).at(j) = direct_distances.at(j).at(i) = calc_direct_dist(i, j);
            }
        }
        for (int k = 0; k < N; ++k) {
            for (int i = 0; i < N; ++i) {
                for (int j = 0; j < N; ++j) {
                    direct_distances.at(i).at(j) = std::min(
                        direct_distances.at(i).at(j), direct_distances.at(i).at(k) + direct_distances.at(k).at(j));
                }
            }
        }

        for (int i = 0; i < M; ++i) {
            C.at(i) = rand_int() % (UB + 1);
            D.at(i) = rand_int() % (UB + 1);
        }


        auto best_mat = gen_best_mat();
        csr_distance_matrix<long long> adj = build_adjacent_info(best_mat, 30);
        auto tsp = SymmetricTSP(best_mat, adj);

        sol = tsp.nearest_neighbor(0);
        dbg(sol.cost);
        three_opt(tsp, sol);
        dbg(sol.cost);
    }

    Matrix gen_best_mat() const {
        auto ret = direct_distances;
        std::vector pqwf(M, std::vector<int>(M));
        for (int p = 0; p < M; ++p) {
            for (int q = 0; q < M; ++q) {
                pqwf.at(p).at(q) = calc_s2s_dist(p, q);
            }
        }
        for (int k = 0; k < M; ++k) {
            for (int p = 0; p < M; ++p) {
                for (int q = 0; q < M; ++q) {
                    pqwf.at(p).at(q) = std::min(pqwf.at(p).at(q), pqwf.at(p).at(k) + pqwf.at(k).at(q));
                }
            }
        }

        for (int i = 0; i < N; ++i) {
            for (int j = 0; j < i; ++j) {
                ret.at(i).at(j) = calc_direct_dist(i, j);
                for (int p = 0; p < M; ++p) {
                    for (int q = 0; q < M; ++q) {
                        ret.at(i).at(j) = std::min<int>(
                            ret.at(i).at(j), calc_p2s_dist(i, p) + pqwf.at(p).at(q) + calc_p2s_dist(j, q));
                    }
                }
                ret.at(j).at(i) = ret.at(i).at(j);
                assert(ret.at(i).at(j) >= 0);
            }
        }
        return Matrix(ret);
    }


    void step(int m, int dmax) {
        const int dx = rand_int() % (dmax * 2 + 1) - dmax;
        const int dy = rand_int() % (dmax * 2 + 1) - dmax;
        C.at(m) += dx;
        D.at(m) += dy;
        auto tmpsol = sol;
        auto best_mat = gen_best_mat();
        csr_distance_matrix<long long> adj = build_adjacent_info(best_mat, 30);
        const auto tsp = SymmetricTSP(best_mat, adj);
        tmpsol.cost = tsp.eval(tmpsol);
        tsp.two_opt(tmpsol);
        if (tmpsol.cost < sol.cost) {
            dbg(tmpsol.cost);
            sol = tmpsol;
        } else {
            C.at(m) -= dx;
            D.at(m) -= dy;
        }
    }

    void finalize() {
        auto best_mat = gen_best_mat();
        csr_distance_matrix<long long> adj = build_adjacent_info(best_mat, 30);
        const auto tsp = SymmetricTSP(best_mat, adj);
        three_opt(tsp, sol);
    }

    void exp_step() {
        const int m = rand_int() % M;
        const int cm = C.at(m), dm = D.at(m);

        int dx = 0, dy = 0;
        while (!dx and !dy) {
            dx = rand_int() % 31 - 15;
            dy = rand_int() % 31 - 15;
        }

        bool init = true;
        while (true) {
            C.at(m) += dx;
            D.at(m) += dy;
            auto tmpsol = sol;
            auto best_mat = gen_best_mat();
            csr_distance_matrix<long long> adj = build_adjacent_info(best_mat, 30);
            const auto tsp = SymmetricTSP(best_mat, adj);
            tmpsol.cost = tsp.eval(tmpsol);
            tsp.two_opt(tmpsol);
            if (tmpsol.cost < sol.cost) {
                dbg(tmpsol.cost);
                sol = tmpsol;
                if (init) {
                    init = false;
                } else {
                    dx *= 2;
                    dy *= 2;
                }
            } else {
                C.at(m) -= dx;
                D.at(m) -= dy;
                dbg(std::make_tuple(m, dx, dy));
                break;
            }
        }
    }

    int score() const { return round(1e9 / (1000 + sqrt(sol.cost))); }


    std::vector<std::pair<int, int>> dump_solution() const {
        shortest_path<long long> graph(N + M);
        for (int i = 0; i < N; ++i) {
            for (int j = i + 1; j < N; ++j) graph.add_bi_edge(i, j, calc_direct_dist(i, j));
            for (int p = 0; p < M; ++p) graph.add_bi_edge(i, N + p, calc_p2s_dist(i, p));
        }
        for (int p = 0; p < M; ++p) {
            for (int q = p + 1; q < M; ++q) graph.add_bi_edge(N + p, N + q, calc_s2s_dist(p, q));
        }

        auto path = sol.path;
        std::rotate(path.begin(), std::find(path.begin(), path.end(), 0), path.end());
        assert(path.front() == 0);
        path.push_back(0);
        dbg(path);
        std::vector<std::pair<int, int>> ret;
        ret.emplace_back(1, 0);
        for (int c = 1; c < (int)path.size(); ++c) {
            int prv = path.at(c - 1), cur = path.at(c);
            graph.solve(prv, cur);
            auto p = graph.retrieve_path(cur);
            for (int d = 1; d < (int)p.size(); ++d) {
                int v = p.at(d);
                if (v < N) ret.emplace_back(1, v);
                else ret.emplace_back(2, v - N);
            }
        }
        return ret;
    }

    std::string build_dump_str() const {
        std::string ret;
        for (int p = 0; p < M; ++p) ret += std::to_string(C.at(p)) + " " + std::to_string(D.at(p)) + "\n";
        auto vec = dump_solution();
        ret += std::to_string(vec.size()) + "\n";
        for (auto [k, v] : vec) {
            ret += std::to_string(k) + " " + std::to_string(v + 1) + "\n";
        }
        return ret;
    }
};

using namespace std;
using lint = long long;
using pint = std::pair<int, int>;
using plint = std::pair<lint, lint>;

struct fast_ios {
    fast_ios() {
        std::cin.tie(nullptr), std::ios::sync_with_stdio(false), std::cout << std::fixed << std::setprecision(20);
    };
} fast_ios_;

int main(int argc, char *argv[]) {
    int X = 0;
    if (argc >= 2) { X = std::stoi(argv[1]); }

    int N, M;
    cin >> N >> M;
    vector<int> A(N), B(N);
    REP(i, N) cin >> A.at(i) >> B.at(i);
    dbg(make_tuple(N, M, A, B));
    dbg(A);
    dbg(B);

    Solver solver(A, B);

    for (int d = 350; d > 0; d -= 8) {
        for (int m = 0; m < M; ++m) solver.step(m, d);
    }
    solver.finalize();

    dbg(solver.sol.cost);
    dbg(solver.score());

    // dump_onlinejudge("solution");
    auto sol = solver.dump_solution();
    dbg(sol);

    dump_onlinejudge(solver.build_dump_str());

    jdump("score", solver.score());
}