/**
 * 
 */

// header {{{
#include <bits/stdc++.h>
using namespace std;

#define CPP_STR(x) CPP_STR_I(x)
#define CPP_CAT(x,y) CPP_CAT_I(x,y)
#define CPP_STR_I(args...) #args
#define CPP_CAT_I(x,y) x ## y

#define SFINAE(pred...) std::enable_if_t<(pred), std::nullptr_t> = nullptr

#define ASSERT(expr...) assert((expr))

using i8  = int8_t;
using u8  = uint8_t;
using i16 = int16_t;
using u16 = uint16_t;
using i32 = int32_t;
using u32 = uint32_t;
using i64 = int64_t;
using u64 = uint64_t;

using f32 = float;
using f64 = double;
using f80 = __float80;
// }}}

constexpr i64 INF  = INT64_C(1'010'000'000'000'000'017);
constexpr f64 FINF = 1e100;

constexpr i64 MOD = INT64_C(1'000'000'007);
//constexpr i64 MOD = INT64_C(998'244'353);

constexpr f64 EPS = 1e-12;

constexpr f64 PI = 3.14159265358979323846;

// util {{{
#define FOR(i, start, end) for(i64 i = (start), CPP_CAT(i,xxxx_end)=(end); i < CPP_CAT(i,xxxx_end); ++i)
#define REP(i, n) FOR(i, 0, n)

#define ALL(f,c,...) (([&](decltype((c)) cccc) { return (f)(std::begin(cccc), std::end(cccc), ## __VA_ARGS__); })(c))

#define LIFT(f) ([](auto&&... args) -> decltype(auto) { return (f)(std::forward<decltype(args)>(args)...); })

template<typename C>
constexpr i64 SIZE(const C& c) noexcept { return static_cast<i64>(c.size()); }

template<typename T, size_t N>
constexpr i64 SIZE(const T (&)[N]) noexcept { return static_cast<i64>(N); }

template<typename T, SFINAE(is_signed<T>::value)>
constexpr T ABS(T x) noexcept {
    return x < 0 ? -x : x;
}

template<typename T>
constexpr i64 CMP(T x, T y) noexcept { return (y<x) - (x<y); }

template<typename T>
constexpr i64 SGN(T x) noexcept { return CMP(x,T(0)); }

template<typename T, typename U, typename Comp=less<>>
constexpr bool chmax(T& xmax, const U& x, Comp comp={}) noexcept {
    if(comp(xmax, x)) {
        xmax = x;
        return true;
    }
    return false;
}

template<typename T, typename U, typename Comp=less<>>
constexpr bool chmin(T& xmin, const U& x, Comp comp={}) noexcept {
    if(comp(x, xmin)) {
        xmin = x;
        return true;
    }
    return false;
}

template<typename BinaryFunc, typename UnaryFunc>
auto ON(BinaryFunc&& bf, UnaryFunc&& uf) {
    return [bf=forward<BinaryFunc>(bf),uf=forward<UnaryFunc>(uf)](const auto& x, const auto& y) {
        return bf(uf(x), uf(y));
    };
}

template<typename F>
auto LT_ON(F&& f) {
    return ON(less<>{}, forward<F>(f));
}

template<typename F>
auto GT_ON(F&& f) {
    return ON(greater<>{}, forward<F>(f));
}

template<typename F>
auto EQ_ON(F&& f) {
    return ON(equal_to<>{}, forward<F>(f));
}

template<typename F>
auto NE_ON(F&& f) {
    return ON(not_equal_to<>{}, forward<F>(f));
}

// tuple {{{
template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) == I)>
void tuple_enumerate(tuple<TS...>&, F&&) {}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) > I)>
void tuple_enumerate(tuple<TS...>& t, F&& f) {
    f(I, get<I>(t));
    tuple_enumerate<I+1>(t, forward<F>(f));
}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) == I)>
void tuple_enumerate(const tuple<TS...>&, F&&) {}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) > I)>
void tuple_enumerate(const tuple<TS...>& t, F&& f) {
    f(I, get<I>(t));
    tuple_enumerate<I+1>(t, forward<F>(f));
}
// }}}

// container {{{
template<typename T> struct is_container : false_type {};
template<typename T, size_t N> struct is_container<array<T,N>> : true_type {};
template<typename... Args> struct is_container<vector<Args...>> : true_type {};
template<typename... Args> struct is_container<deque<Args...>> : true_type {};
template<typename... Args> struct is_container<list<Args...>> : true_type {};
template<typename... Args> struct is_container<forward_list<Args...>> : true_type {};
template<typename... Args> struct is_container<set<Args...>> : true_type {};
template<typename... Args> struct is_container<multiset<Args...>> : true_type {};
template<typename... Args> struct is_container<unordered_set<Args...>> : true_type {};
template<typename... Args> struct is_container<unordered_multiset<Args...>> : true_type {};
template<typename... Args> struct is_container<map<Args...>> : true_type {};
template<typename... Args> struct is_container<multimap<Args...>> : true_type {};
template<typename... Args> struct is_container<unordered_map<Args...>> : true_type {};
template<typename... Args> struct is_container<unordered_multimap<Args...>> : true_type {};

template<typename T, typename Enable=void>
struct ProconHash {
    size_t operator()(const T& x) const noexcept {
        return hash<T>{}(x);
    }
};

template<typename T>
size_t procon_hash_value(const T& x) noexcept {
    return ProconHash<T>{}(x);
}

size_t procon_hash_combine(size_t h1, size_t h2) noexcept {
    constexpr size_t M = UINT64_C(0xc6a4a7935bd1e995);
    constexpr int    R = 47;

    h2 *= M;
    h2 ^= h2 >> R;
    h2 *= M;

    h1 ^= h2;
    h1 *= M;

    h1 += 0xe6546b64;

    return h1;
}

template<typename T1, typename T2>
struct ProconHash<pair<T1,T2>> {
    size_t operator()(const pair<T1,T2>& p) const noexcept {
        size_t h1 = procon_hash_value(p.first);
        size_t h2 = procon_hash_value(p.second);
        return procon_hash_combine(h1, h2);
    }
};

template<typename... TS>
struct ProconHash<tuple<TS...>> {
    size_t operator()(const tuple<TS...>& t) const noexcept {
        size_t h = 0;
        tuple_enumerate(t, [&h](i64, const auto& e) {
            h = procon_hash_combine(h, procon_hash_value(e));
        });
        return h;
    }
};

template<typename C>
struct ProconHash<C,enable_if_t<is_container<C>::value>> {
    size_t operator()(const C& c) const noexcept {
        size_t h = 0;
        for(const auto& e : c)
            h = procon_hash_combine(h, procon_hash_value(e));
        return h;
    }
};

template<typename T, typename Hash=ProconHash<T>, typename Eq=equal_to<T>>
using HashSet = unordered_set<T,Hash,Eq>;
template<typename K, typename V, typename Hash=ProconHash<K>, typename Eq=equal_to<K>>
using HashMap = unordered_map<K,V,Hash,Eq>;
template<typename T, typename Hash=ProconHash<T>, typename Eq=equal_to<T>>
using HashMultiset = unordered_multiset<T,Hash,Eq>;
template<typename K, typename V, typename Hash=ProconHash<K>, typename Eq=equal_to<K>>
using HashMultimap = unordered_multimap<K,V,Hash,Eq>;

template<typename T>
auto vec_make(i64 n, T x) {
    return vector<T>(n, x);
}

template<typename T, typename... Args, SFINAE(sizeof...(Args) >= 2)>
auto vec_make(i64 n, Args... args) {
    auto inner = vec_make<T>(args...);
    return vector<decltype(inner)>(n, inner);
}

template<typename T>
auto vec_reserve(i64 cap) {
    vector<T> res;
    res.reserve(cap);
    return res;
}

template<typename T=i64>
auto vec_iota(i64 n, T init={}) {
    vector<i64> res(n);
    ALL(iota, res, init);
    return res;
}

template<typename T, typename Comp, typename Cont=vector<T>>
auto priority_queue_make(const Comp& comp, Cont&& cont={}) {
    return priority_queue<T,remove_reference_t<Cont>,Comp>(comp, forward<Cont>(cont));
}

template<typename T, typename Comp>
auto priority_queue_reserve(const Comp& comp, i64 cap) {
    return priority_queue<T,vector<T>,Comp>(comp, vec_reserve<T>(cap));
}

template<typename T, size_t N, size_t... NS>
struct ArrayType {
    using type = array<typename ArrayType<T,NS...>::type,N>;
};

template<typename T, size_t N>
struct ArrayType<T,N> {
    using type = array<T,N>;
};

template<typename T, size_t... NS>
using Array = typename ArrayType<T,NS...>::type;

template<typename T, size_t N>
T& array_at(Array<T,N>& ary, i64 i) {
    return ary[i];
}

template<typename T, size_t N, size_t... NS, typename... Args>
T& array_at(Array<T,N,NS...>& ary, i64 i, Args... args) {
    return array_at<T,NS...>(ary[i], args...);
}

template<typename T, size_t N>
const T& array_at(const Array<T,N>& ary, i64 i) {
    return ary[i];
}

template<typename T, size_t N, size_t... NS, typename... Args>
const T& array_at(const Array<T,N,NS...>& ary, i64 i, Args... args) {
    return array_at<T,NS...>(ary[i], args...);
}

template<typename T, typename C>
T POP(stack<T,C>& stk) {
    T x = stk.top(); stk.pop();
    return x;
}

template<typename T, typename C>
T POP(queue<T,C>& que) {
    T x = que.front(); que.pop();
    return x;
}

template<typename T, typename C, typename Comp>
T POP(priority_queue<T,C,Comp>& que) {
    T x = que.top(); que.pop();
    return x;
}
// }}}

// fixpoint {{{
template<typename F>
class FixPoint {
public:
    explicit constexpr FixPoint(F&& f) : f_(forward<F>(f)) {}

    template<typename... Args>
    constexpr decltype(auto) operator()(Args&&... args) const {
        return f_(*this, forward<Args>(args)...);
    }

private:
    F f_;
};

template<typename F>
constexpr decltype(auto) FIX(F&& f) {
    return FixPoint<F>(forward<F>(f));
}

template<typename F, size_t... NS>
class FixPointMemo {
public:
    explicit FixPointMemo(F&& f) : f_(forward<F>(f)) {}

    template<typename... Args>
    decltype(auto) operator()(Args... args) const {
        using R = decltype(f_(*this,args...));
        static Array<bool,NS...> done {};
        static Array<R,NS...>    memo;

        if(!array_at<bool,NS...>(done,args...)) {
            array_at<R,NS...>(memo,args...) = f_(*this,args...);
            array_at<bool,NS...>(done,args...) = true;
        }
        return array_at<R,NS...>(memo,args...);
    }

private:
    F f_;
};

template<size_t... NS, typename F>
decltype(auto) FIXMEMO(F&& f) {
    return FixPointMemo<F,NS...>(forward<F>(f));
}
// }}}

// math {{{
/*constexpr*/ i64 GCD(i64 a, i64 b) noexcept {
    /*constexpr*/ auto f_gcd = FIX([](auto&& self, i64 aa, i64 bb) {
        if(bb == 0) return aa;
        return self(bb, aa%bb);
    });
    return f_gcd(ABS(a), ABS(b));
}

/*constexpr*/ i64 LCM(i64 a, i64 b) noexcept {
    ASSERT(a != 0 && b != 0);
    /*constexpr*/ auto f_gcd = FIX([](auto&& self, i64 aa, i64 bb) {
        if(bb == 0) return aa;
        return self(bb, aa%bb);
    });
    a = ABS(a);
    b = ABS(b);
    return a / f_gcd(a,b) * b;
}

/*constexpr*/ tuple<i64,i64,i64> EXTGCD(i64 a, i64 b) noexcept {
    /*constexpr*/ auto impl = FIX([](auto&& self, i64 aa, i64 bb) -> tuple<i64,i64,i64> {
        if(bb == 0) return make_tuple(aa, 1, 0);
        i64 g,x,y; tie(g,x,y) = self(bb, aa%bb);
        return make_tuple(g, y, x-(aa/bb)*y);
    });
    i64 g,x,y; tie(g,x,y) = impl(ABS(a), ABS(b));
    x *= SGN(a);
    y *= SGN(b);
    return make_tuple(g, x, y);
}
// }}}

// string {{{
auto str_reserve(i64 cap) {
    string res;
    res.reserve(cap);
    return res;
}
// }}}

// input {{{
template<typename T, typename Enable=void>
struct Scan {
    static T scan(istream& in) {
        T res;
        in >> res;
        return res;
    }
};

template<typename T, typename Enable=void>
struct Scan1;

template<typename T>
struct Scan1<T,enable_if_t<is_integral<T>::value && !is_same<T,bool>::value>> {
    static T scan1(istream& in) {
        return Scan<T>::scan(in) - 1;
    }
};

template<typename T1, typename T2>
struct Scan<pair<T1,T2>> {
    static pair<T1,T2> scan(istream& in) {
        T1 x = Scan<T1>::scan(in);
        T2 y = Scan<T2>::scan(in);
        return {x,y};
    }
};

template<typename T1, typename T2>
struct Scan1<pair<T1,T2>> {
    static pair<T1,T2> scan1(istream& in) {
        T1 x = Scan1<T1>::scan1(in);
        T2 y = Scan1<T2>::scan1(in);
        return {x,y};
    }
};

template<typename T>
tuple<T> tuple_scan_impl(istream& in) {
    return make_tuple(Scan<T>::scan(in));
}

template<typename T, typename... TS, SFINAE(sizeof...(TS) > 0)>
tuple<T,TS...> tuple_scan_impl(istream& in) {
    auto head = make_tuple(Scan<T>::scan(in));
    return tuple_cat(head, tuple_scan_impl<TS...>(in));
}

template<typename... TS>
struct Scan<tuple<TS...>> {
    static tuple<TS...> scan(istream& in) {
        return tuple_scan_impl<TS...>(in);
    }
};

template<typename T>
tuple<T> tuple_scan1_impl(istream& in) {
    return make_tuple(Scan1<T>::scan1(in));
}

template<typename T, typename... TS, SFINAE(sizeof...(TS) > 0)>
tuple<T,TS...> tuple_scan1_impl(istream& in) {
    auto head = make_tuple(Scan1<T>::scan1(in));
    return tuple_cat(head, tuple_scan1_impl<TS...>(in));
}

template<typename... TS>
struct Scan1<tuple<TS...>> {
    static tuple<TS...> scan1(istream& in) {
        return tuple_scan1_impl<TS...>(in);
    }
};

template<typename T=i64>
T RD() {
    return Scan<T>::scan(cin);
}

template<typename T=i64>
T RD1() {
    return Scan1<T>::scan1(cin);
}

template<typename T=i64>
auto RD_VEC(i64 n) {
    auto res = vec_reserve<T>(n);
    REP(_, n) {
        res.emplace_back(RD<T>());
    }
    return res;
}

template<typename T=i64>
auto RD1_VEC(i64 n) {
    auto res = vec_reserve<T>(n);
    REP(_, n) {
        res.emplace_back(RD1<T>());
    }
    return res;
}

template<typename T=i64>
auto RD_VEC2(i64 h, i64 w) {
    auto res = vec_reserve<vector<T>>(h);
    REP(_, h) {
        res.emplace_back(RD_VEC<T>(w));
    }
    return res;
}

template<typename T=i64>
auto RD1_VEC2(i64 h, i64 w) {
    auto res = vec_reserve<vector<T>>(h);
    REP(_, h) {
        res.emplace_back(RD1_VEC<T>(w));
    }
    return res;
}
// }}}

// output {{{
template<typename T, typename Enable=void>
struct Fmt {
    static void fmt(ostream& out, const T& x) { out << x; }
};

template<typename T>
void fmt_write(ostream& out, const T& x) {
    Fmt<T>::fmt(out, x);
}

template<typename... TS>
struct Fmt<tuple<TS...>> {
    static void fmt(ostream& out, const tuple<TS...>& t) {
        tuple_enumerate(t, [&out](i64 i, const auto& e) {
            if(i != 0) out << ' ';
            fmt_write(out, e);
        });
    }
};

template<typename T1, typename T2>
struct Fmt<pair<T1,T2>> {
    static void fmt(ostream& out, const pair<T1,T2>& p) {
        return fmt_write(out, make_tuple(p.first,p.second));
    }
};

template<typename C>
struct Fmt<C,enable_if_t<is_container<C>::value>> {
    static void fmt(ostream& out, const C& c) {
        for(auto it = begin(c); it != end(c); ++it) {
            if(it != begin(c)) out << ' ';
            fmt_write(out, *it);
        }
    }
};

void PRINT() {}

template<typename T, typename... TS>
void PRINT(const T& x, const TS&... args) {
    fmt_write(cout, x);
    if(sizeof...(args) > 0) {
        cout << ' ';
        PRINT(args...);
    }
}

template<typename... TS>
void PRINTLN(const TS&... args) {
    PRINT(args...);
    cout << '\n';
}
// }}}

// debug {{{
template<typename T, typename Enable=void>
struct Dbg {
    static void dbg(ostream& out, const T& x) { out << x; }
};

template<typename T>
void dbg_write(ostream& out, const T& x) {
    Dbg<T>::dbg(out, x);
}

template<>
struct Dbg<i64> {
    static void dbg(ostream& out, i64 x) {
        if(x == INF)
            out << "INF";
        else if(x == -INF)
            out << "-INF";
        else
            out << x;
    }
};

template<>
struct Dbg<f64> {
    static void dbg(ostream& out, f64 x) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
        if(x == FINF)
            out << "FINF";
        else if(x == -FINF)
            out << "-FINF";
        else
            out << x;
#pragma GCC diagnostic pop
    }
};

template<typename T, size_t N>
struct Dbg<T[N]> {
    static void dbg(ostream& out, const T (&ary)[N]) {
        out << "[";
        REP(i, N) {
            if(i != 0) out << ",";
            dbg_write(out, ary[i]);
        }
        out << "]";
    }
};

template<size_t N>
struct Dbg<char[N]> {
    static void dbg(ostream& out, const char (&s)[N]) {
        out << s;
    }
};

template<typename... TS>
struct Dbg<tuple<TS...>> {
    static void dbg(ostream& out, const tuple<TS...>& t) {
        out << "(";
        tuple_enumerate(t, [&out](i64 i, const auto& e) {
            if(i != 0) out << ",";
            dbg_write(out, e);
        });
        out << ")";
    }
};

template<typename T1, typename T2>
struct Dbg<pair<T1,T2>> {
    static void dbg(ostream& out, const pair<T1,T2>& p) {
        return dbg_write(out, make_tuple(p.first,p.second));
    }
};

template<typename C>
struct Dbg<C,enable_if_t<is_container<C>::value>> {
    static void dbg(ostream& out, const C& c) {
        out << "[";
        for(auto it = begin(c); it != end(c); ++it) {
            if(it != begin(c)) out << ",";
            dbg_write(out, *it);
        }
        out << "]";
    }
};

template<typename T, typename C>
struct Dbg<stack<T,C>> {
    static void dbg(ostream& out, stack<T,C> stk) {
        out << "[";
        while(!stk.empty()) {
            dbg_write(out,stk.top()); stk.pop();
            if(!stk.empty()) out << ",";
        }
        out << "]";
    }
};

template<typename T, typename C>
struct Dbg<queue<T,C>> {
    static void dbg(ostream& out, queue<T,C> que) {
        out << "[";
        while(!que.empty()) {
            dbg_write(out,que.front()); que.pop();
            if(!que.empty()) out << ",";
        }
        out << "]";
    }
};

template<typename T, typename C, typename Comp>
struct Dbg<priority_queue<T,C,Comp>> {
    static void dbg(ostream& out, priority_queue<T,C,Comp> que) {
        out << "[";
        while(!que.empty()) {
            dbg_write(out,que.top()); que.pop();
            if(!que.empty()) out << ",";
        }
        out << "]";
    }
};

template<typename T>
void DBG_IMPL(i64 line, const char* expr, const T& value) {
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    dbg_write(cerr, value);
    cerr << "\n";
}

void DBG_IMPL_HELPER() {}

template<typename T, typename... TS>
void DBG_IMPL_HELPER(const T& x, const TS&... args) {
    dbg_write(cerr, x);
    if(sizeof...(args) > 0) {
        cerr << ",";
        DBG_IMPL_HELPER(args...);
    }
}

template<typename... TS>
void DBG_IMPL(i64 line, const char* expr, const TS&... value) {
    cerr << "[L " << line << "]: ";
    cerr << "(" << expr << ") = (";
    DBG_IMPL_HELPER(value...);
    cerr << ")\n";
}

template<size_t N, typename T, SFINAE(rank<T>::value == 0)>
void DBG_DP_IMPL_HELPER(ostream& out, const T& x, const array<i64,N>&, const array<i64,N>&) {
    dbg_write(out, x);
}

template<size_t N, typename T, SFINAE(rank<T>::value > 0)>
void DBG_DP_IMPL_HELPER(ostream& out, const T& x, const array<i64,N>& sizes, const array<i64,N>& offs) {
    i64 k   = N - rank<T>::value;
    i64 off = offs[k];
    i64 siz = sizes[k];
    if(siz == 0) siz = extent<T>::value - off;

    out << "[";
    FOR(i, off, off+siz) {
        if(i != off) out << ",";
        DBG_DP_IMPL_HELPER(out, x[i], sizes, offs);
    }
    out << "]";
}

template<typename T, SFINAE(rank<T>::value > 0)>
void DBG_DP_IMPL(i64 line, const char* expr, const T& dp,
                 const array<i64,rank<T>::value>& sizes={},
                 const array<i64,rank<T>::value>& offs={})
{
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    DBG_DP_IMPL_HELPER<rank<T>::value>(cerr, dp, sizes, offs);
    cerr << "\n";
}

template<typename T>
void DBG_GRID_IMPL(i64 line, const char* expr, const vector<T>& grid) {
    cerr << "[L " << line << "]: ";
    cerr << expr << ":\n";
    for(const auto& row : grid) {
        dbg_write(cerr, row);
        cerr << "\n";
    }
    cerr << "\n";
}

#ifdef PROCON_LOCAL
    #define DBG(args...) DBG_IMPL(__LINE__, CPP_STR_I(args), args)
    #define DBG_DP(args...) DBG_DP_IMPL(__LINE__, CPP_STR_I(args), args)
    #define DBG_GRID(args...) DBG_GRID_IMPL(__LINE__, CPP_STR_I(args), args)
#else
    #define DBG(args...)
    #define DBG_DP(args...)
    #define DBG_GRID(args...)
#endif
// }}}

// modint {{{
template<typename Mod>
class ModIntT {
private:
    i64 v_;  // [0,Mod::value)

    static i64 mod() { return Mod::value; }

    static i64 normalize(i64 x) {
        i64 res = x % mod();
        if(res < 0) res += mod();
        return res;
    }

public:
    ModIntT() : v_(0) {}
    ModIntT(i64 v) : v_(normalize(v)) {}

    explicit operator i64() const { return v_; }

    ModIntT operator-() const { return ModIntT(-v_); }

    ModIntT& operator+=(ModIntT rhs) {
        v_ = normalize(v_ + rhs.v_);
        return *this;
    }
    ModIntT& operator-=(ModIntT rhs) {
        v_ = normalize(v_ - rhs.v_);
        return *this;
    }
    ModIntT& operator*=(ModIntT rhs) {
        v_ = normalize(v_ * rhs.v_);
        return *this;
    }

    ModIntT& operator++() { return *this += 1; }
    ModIntT& operator--() { return *this -= 1; }
    ModIntT operator++(int) { return exchange(*this, *this+1); }
    ModIntT operator--(int) { return exchange(*this, *this-1); }

    ModIntT inv() const {
        i64 g,x; tie(g,x,ignore) = EXTGCD(v_, mod());
        ASSERT(g == 1);
        return ModIntT(x);
    }
};

template<typename Mod>
ModIntT<Mod> operator+(ModIntT<Mod> lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(lhs) += rhs; }
template<typename Mod>
ModIntT<Mod> operator+(ModIntT<Mod> lhs, i64 rhs) { return ModIntT<Mod>(lhs) += rhs; }
template<typename Mod>
ModIntT<Mod> operator+(i64 lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(rhs) += lhs; }
template<typename Mod>
ModIntT<Mod> operator-(ModIntT<Mod> lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(lhs) -= rhs; }
template<typename Mod>
ModIntT<Mod> operator-(ModIntT<Mod> lhs, i64 rhs) { return ModIntT<Mod>(lhs) -= rhs; }
template<typename Mod>
ModIntT<Mod> operator-(i64 lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(rhs) -= lhs; }
template<typename Mod>
ModIntT<Mod> operator*(ModIntT<Mod> lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(lhs) *= rhs; }
template<typename Mod>
ModIntT<Mod> operator*(ModIntT<Mod> lhs, i64 rhs) { return ModIntT<Mod>(lhs) *= rhs; }
template<typename Mod>
ModIntT<Mod> operator*(i64 lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(rhs) *= lhs; }

template<typename Mod>
bool operator==(ModIntT<Mod> lhs, ModIntT<Mod> rhs) { return i64(lhs) == i64(rhs); }
template<typename Mod>
bool operator==(ModIntT<Mod> lhs, i64 rhs) { return lhs == ModIntT<Mod>(rhs); }
template<typename Mod>
bool operator==(i64 lhs, ModIntT<Mod> rhs) { return ModIntT<Mod>(lhs) == rhs; }
template<typename Mod>
bool operator!=(ModIntT<Mod> lhs, ModIntT<Mod> rhs) { return !(lhs == rhs); }
template<typename Mod>
bool operator!=(ModIntT<Mod> lhs, i64 rhs) { return !(lhs == rhs); }
template<typename Mod>
bool operator!=(i64 lhs, ModIntT<Mod> rhs) { return !(lhs == rhs); }

template<typename Mod>
struct ProconHash<ModIntT<Mod>> {
    size_t operator()(ModIntT<Mod> x) const noexcept {
        return procon_hash_value(i64(x));
    }
};

template<typename Mod>
struct Scan<ModIntT<Mod>> {
    static ModIntT<Mod> scan(istream& in) {
        i64 v = Scan<i64>::scan(in);
        return ModIntT<Mod>(v);
    }
};

template<typename Mod>
struct Fmt<ModIntT<Mod>> {
    static void fmt(ostream& out, ModIntT<Mod> x) {
        fmt_write(out, i64(x));
    }
};

template<typename Mod>
struct Dbg<ModIntT<Mod>> {
    static void dbg(ostream& out, ModIntT<Mod> x) {
        dbg_write(out, i64(x));
    }
};

template<i64 M>
using ModIntC = ModIntT<integral_constant<i64,M>>;

using ModInt = ModIntC<MOD>;
// }}}
// }}}

// init {{{
struct ProconInit {
    static constexpr int IOS_PREC = 15;
    static constexpr bool AUTOFLUSH = false;

    ProconInit() {
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cin.exceptions(ios::failbit | ios::badbit);
        cout << fixed << setprecision(IOS_PREC);
#ifdef PROCON_LOCAL
        cerr << fixed << setprecision(2);
#endif
        if(AUTOFLUSH)
            cout << unitbuf;
    }
} PROCON_INIT;
// }}}

struct UnionFind {
    vector<i64> ps_;  // 親ノード。ただし根の場合 -(集合の要素数)
    i64 size_;

    explicit UnionFind(i64 n) : ps_(n,-1), size_(n) {}

    i64 size() const { return size_; }

    i64 root(i64 x) {
        i64 p = ps_[x];
        if(p < 0) return x;
        return ps_[x] = root(p);
    }

    i64 group_size(i64 x) {
        return -ps_[root(x)];
    }

    void unite(i64 x, i64 y) {
        i64 rx = root(x);
        i64 ry = root(y);
        if(rx == ry) return;

        // 要素数が大きい方を根とする
        i64 kx = -ps_[rx];
        i64 ky = -ps_[ry];
        if(kx < ky) swap(rx, ry);
        ps_[rx] = -(kx + ky);
        ps_[ry] = rx;

        --size_;
    }
};

//--------------------------------------------------------------------

auto coord_compress(vector<i64> xs) {
    ALL(sort, xs);
    xs.erase(ALL(unique,xs), end(xs));

    HashMap<i64,i64> comp;
    REP(i, SIZE(xs)) {
        comp[xs[i]] = i;
    }
    return comp;
}

void solve(i64 N, i64 M) {
    auto H = RD_VEC(N);
    auto hcomp = coord_compress(H);
    vector<vector<i64>> timeline(SIZE(hcomp));
    REP(v, N) {
        i64 h = H[v];
        timeline[hcomp[h]].emplace_back(v);
    }
    DBG(timeline);

    auto es = vec_reserve<tuple<i64,i64,i64>>(M);
    REP(_, M) {
        i64 a = RD1();
        i64 b = RD1();
        i64 c = RD();
        es.emplace_back(c, a, b);
    }
    ALL(sort, es);
    DBG(es);

    // 橋を全て架けたとして、*最初に*グラフが非連結になる時刻を求める
    // (時刻が進むにつれ連結になったり非連結になったりすることに注意)
    // 時刻の逆順に処理
    i64 i_discon = SIZE(timeline);
    {
        UnionFind uf(N);
        i64 k = 0;
        vector<bool> exists(N, false);
        for(i64 i = SIZE(timeline)-1; i >= 0; --i) {
            const auto& vs = timeline[i];
            k += SIZE(vs);
            for(i64 v : vs)
                exists[v] = true;

            for(const auto& e : es) {
                i64 v,w; tie(ignore,v,w) = e;
                if(!exists[v] || !exists[w]) continue;
                uf.unite(v, w);
            }

            if(SIZE(uf) > N-k+1)
                i_discon = i;
        }
    }
    DBG(i_discon);
    // 最初から非連結なら何もしない
    if(i_discon == 0) {
        PRINTLN(0);
        return;
    }

    // i_discon の1つ手前から逆順に見ていく
    // 各時刻で最小全域木を構築(Kruskal)
    vector<bool> exists(N, false);
    for(i64 i = SIZE(timeline)-1; i >= i_discon; --i) {
        for(i64 v : timeline[i])
            exists[v] = true;
    }
    UnionFind uf(N);
    i64 ans = 0;
    for(i64 i = i_discon-1; i >= 0; --i) {
        for(i64 v : timeline[i])
            exists[v] = true;

        for(const auto& e : es) {
            i64 cost,v,w; tie(cost,v,w) = e;
            if(!exists[v] || !exists[w]) continue;
            if(uf.root(v) == uf.root(w)) continue;
            uf.unite(v, w);
            ans += cost;
        }
    }

    PRINTLN(ans);
}

signed main() {
    for(;;) {
        i64 N = RD();
        i64 M = RD();
        if(N == 0 && M == 0) break;

        solve(N, M);
    }

    return 0;
}



Measured memory_bytes: 3676