/**
 * 
 */

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

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;
#ifdef __SIZEOF_INT128__
using i128 = __int128;
using u128 = unsigned __int128;
#endif

using f32  = float;
using f64  = double;
using f80  = __float80;
using f128 = __float128;
// }}}

constexpr i64 INF = 1'010'000'000'000'000'000LL;

constexpr i64 MOD = 1'000'000'007LL;

constexpr f64 EPS = 1e-12;

constexpr f64 PI = 3.14159265358979323846;

// {{{ util
#define FOR(i, start, end) for(i64 i = (start), i##_end=(end); i < i##_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 SLICE(f,c,l,r,...) (([&](decltype((c)) cccc, decltype((l)) llll, decltype((r)) rrrr) {\
    auto iiii = llll <= rrrr ? std::begin(cccc)+llll : std::end(cccc);\
    auto jjjj = llll <= rrrr ? std::begin(cccc)+rrrr : std::end(cccc);\
    return (f)(iiii, jjjj, ## __VA_ARGS__);\
})(c,l,r))

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

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:
    const F f_;
};

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

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

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

bool is_odd (i64 x) { return x % 2 != 0; }
bool is_even(i64 x) { return x % 2 == 0; }

template<typename T> i64 cmp(T x, T y) { return (y<x) - (x<y); }
template<typename T> i64 sgn(T x) { return cmp(x, T(0)); }

// Haskell の divMod と同じ
pair<i64,i64> divmod(i64 a, i64 b) {
    i64 q = a / b;
    i64 r = a % b;
    if((b>0 && r<0) || (b<0 && r>0)) {
        --q;
        r += b;
    }
    return {q,r};
}

i64 div_ceil(i64 a, i64 b) {
    i64 q = a / b;
    i64 r = a % b;
    if((b>0 && r>0) || (b<0 && r<0))
        ++q;
    return q;
}

i64 div_floor(i64 a, i64 b) {
    return divmod(a,b).first;
}

i64 modulo(i64 a, i64 b) {
    return divmod(a,b).second;
}

bool feq(f64 x, f64 y, f64 eps=EPS) {
    return fabs(x-y) < eps;
}

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

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

template<typename InputIt>
auto SUM(InputIt first, InputIt last) {
    using T = typename iterator_traits<InputIt>::value_type;
    return accumulate(first, last, T());
}

template<typename ForwardIt, typename UnaryOperation>
ForwardIt transform_self(ForwardIt first, ForwardIt last, UnaryOperation op) {
    return transform(first, last, first, op);
}

template<typename C>
void UNIQ(C& c) {
    c.erase(ALL(unique,c), end(c));
}

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

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

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

char digit_chr(i64 n) {
    return static_cast<char>('0' + n);
}

i64 digit_ord(char c) {
    return c - '0';
}

char lower_chr(i64 n) {
    return static_cast<char>('a' + n);
}

i64 lower_ord(char c) {
    return c - 'a';
}

char upper_chr(i64 n) {
    return static_cast<char>('A' + n);
}

i64 upper_ord(char c) {
    return c - 'A';
}

// 出力は operator<< を直接使わず、このテンプレート経由で行う
// 提出用出力とデバッグ用出力を分けるため
template<typename T>
struct Formatter {
    static ostream& write_str(ostream& out, const T& x)  { return out << x; }
    static ostream& write_repr(ostream& out, const T& x) { return out << x; }
};

template<typename T>
ostream& WRITE_STR(ostream& out, const T& x) {
    return Formatter<T>::write_str(out, x);
}

template<typename T>
ostream& WRITE_REPR(ostream& out, const T& x) {
    return Formatter<T>::write_repr(out, x);
}

template<typename InputIt>
ostream& WRITE_JOIN_STR(ostream& out, InputIt first, InputIt last, const string& sep) {
    while(first != last) {
        WRITE_STR(out, *first++);
        if(first != last)
            out << sep;
    }
    return out;
}

template<typename InputIt>
ostream& WRITE_JOIN_REPR(ostream& out, InputIt first, InputIt last, const string& sep) {
    while(first != last) {
        WRITE_REPR(out, *first++);
        if(first != last)
            out << sep;
    }
    return out;
}

template<typename InputIt>
ostream& WRITE_RANGE_STR(ostream& out, InputIt first, InputIt last) {
    return WRITE_JOIN_STR(out, first, last, " ");
}

template<typename InputIt>
ostream& WRITE_RANGE_REPR(ostream& out, InputIt first, InputIt last) {
    out << "[";
    WRITE_JOIN_REPR(out, first, last, ", ");
    out << "]";
    return out;
}

template<typename T>
void FROM_STR(const string& s, T& x) {
    istringstream in(s);
    in >> x;
}

template<typename T>
string TO_STR(const T& x) {
    ostringstream out;
    WRITE_STR(out, x);
    return out.str();
}

template<typename T>
string TO_REPR(const T& x) {
    ostringstream out;
    WRITE_REPR(out, x);
    return out.str();
}

template<typename InputIt>
string RANGE_TO_STR(InputIt first, InputIt last) {
    ostringstream out;
    WRITE_RANGE_STR(out, first, last);
    return out.str();
}

template<typename InputIt>
string RANGE_TO_REPR(InputIt first, InputIt last) {
    ostringstream out;
    WRITE_RANGE_REPR(out, first, last);
    return out.str();
}

template<typename InputIt>
string JOIN(InputIt first, InputIt last, const string& sep) {
    ostringstream out;
    WRITE_JOIN_STR(out, first, last, sep);
    return out.str();
}

template<typename T>
struct Formatter<vector<T>> {
    static ostream& write_str(ostream& out, const vector<T>& v) {
        return WRITE_RANGE_STR(out, begin(v), end(v));
    }
    static ostream& write_repr(ostream& out, const vector<T>& v) {
        out << "vector";
        return WRITE_RANGE_REPR(out, begin(v), end(v));
    }
};

template<typename T1, typename T2>
struct Formatter<pair<T1,T2>> {
    static ostream& write_str(ostream& out, const pair<T1,T2>& p) {
        WRITE_STR(out, p.first);
        out << ' ';
        WRITE_STR(out, p.second);
        return out;
    }
    static ostream& write_repr(ostream& out, const pair<T1,T2>& p) {
        out << "(";
        WRITE_REPR(out, p.first);
        out << ",";
        WRITE_REPR(out, p.second);
        out << ")";
        return out;
    }
};

template<typename... TS>
struct Formatter<tuple<TS...>> {
    template<size_t I=0, enable_if_t<I == sizeof...(TS), nullptr_t> = nullptr>
    static ostream& write_str_impl(ostream& out, const tuple<TS...>&) {
        return out;
    }
    template<size_t I=0, enable_if_t<I < sizeof...(TS), nullptr_t> = nullptr>
    static ostream& write_str_impl(ostream& out, const tuple<TS...>& t) {
        if(I != 0) out << ' ';
        WRITE_STR(out, get<I>(t));
        return write_str_impl<I+1>(out, t);
    }

    template<size_t I=0, enable_if_t<I == sizeof...(TS), nullptr_t> = nullptr>
    static ostream& write_repr_impl(ostream& out, const tuple<TS...>&) {
        if(sizeof...(TS) == 0) out << "(";
        return out << ")";
    }
    template<size_t I=0, enable_if_t<I < sizeof...(TS), nullptr_t> = nullptr>
    static ostream& write_repr_impl(ostream& out, const tuple<TS...>& t) {
        if(I == 0)
            out << "(";
        else
            out << ",";
        WRITE_REPR(out, get<I>(t));
        return write_repr_impl<I+1>(out, t);
    }

    static ostream& write_str(ostream& out, const tuple<TS...>& t) {
        return write_str_impl(out, t);
    }
    static ostream& write_repr(ostream& out, const tuple<TS...>& t) {
        return write_repr_impl(out, t);
    }
};

template<typename T>
void RD(T& x) {
    cin >> x;
#ifdef PROCON_LOCAL
    assert(cin);
#endif
}

template<typename T>
void RD(vector<T>& v, i64 n) {
    v.reserve(n);
    REP(_, n) {
        T e; RD(e);
        v.emplace_back(e);
    }
}

void PRINT() {}

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

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

[[noreturn]] void EXIT() {
#ifdef PROCON_LOCAL
    cerr.flush();
#endif
    cout.flush();
    _Exit(0);
}

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

template<typename T, size_t N>
void DBG_ARRAY_IMPL(i64 line, const char* expr, const T (&ary)[N]) {
#ifdef PROCON_LOCAL
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    WRITE_RANGE_REPR(cerr, begin(ary), end(ary));
    cerr << "\n";
#endif
}

template<typename InputIt>
void DBG_RANGE_IMPL(i64 line, const char* expr1, const char* expr2, InputIt first, InputIt last) {
#ifdef PROCON_LOCAL
    cerr << "[L " << line << "]: ";
    cerr << expr1 << "," << expr2 << " = ";
    WRITE_RANGE_REPR(cerr, first, last);
    cerr << "\n";
#endif
}

#define DBG(expr) DBG_IMPL(__LINE__, #expr, (expr))
#define DBG_ARRAY(expr) DBG_ARRAY_IMPL(__LINE__, #expr, (expr))
#define DBG_RANGE(first,last) DBG_RANGE_IMPL(__LINE__, #first, #last, (first), (last))
// }}}

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

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

// {{{ container

// {{{ BoolArray
class BoolArray {
public:
    using value_type      = bool;
    using reference       = value_type&;
    using const_reference = const value_type&;
    using iterator        = value_type*;
    using const_iterator  = const value_type*;
    using difference_type = ptrdiff_t;
    using size_type       = size_t;

    using reverse_iterator       = std::reverse_iterator<iterator>;
    using const_reverse_iterator = std::reverse_iterator<const_iterator>;

    BoolArray() : BoolArray(0) {}
    explicit BoolArray(size_t n) : BoolArray(n,false) {}
    BoolArray(size_t n, bool value) : size_(n), data_(new bool[n]) {
        ALL(fill, *this, value);
    }

    BoolArray(initializer_list<bool> init) : size_(init.size()), data_(new bool[size_]) {
        ALL(copy, init, begin());
    }

    template<typename InputIt>
    BoolArray(InputIt first, InputIt last) {
        deque<bool> tmp(first, last);
        size_ = tmp.size();
        data_ = new bool[size_];
        ALL(copy, tmp, begin());
    }

    BoolArray(const BoolArray& other) : size_(other.size_), data_(new bool[size_]) {
        ALL(copy, other, begin());
    }

    BoolArray(BoolArray&& other) : size_(other.size_), data_(other.data_) {
        other.data_ = nullptr;
    }

    BoolArray& operator=(const BoolArray& other) {
        if(this == &other) return *this;
        if(!data_ || size_ < other.size_) {
            delete[] data_;
            data_ = new bool[other.size_];
        }
        size_ = other.size_;
        ALL(copy, other, begin());
        return *this;
    }

    BoolArray& operator=(BoolArray&& other) {
        if(this == &other) return *this;
        size_ = other.size_;
        data_ = other.data_;
        other.data_ = nullptr;
    }

    BoolArray& operator=(initializer_list<bool> init) {
        if(!data_ || size_ < init.size()) {
            delete[] data_;
            data_ = new bool[init.size()];
        }
        size_ = init.size();
        ALL(copy, init, begin());
        return *this;
    }

    void swap(BoolArray& other) noexcept {
        std::swap(size_, other.size_);
        std::swap(data_, other.data_);
    }

    ~BoolArray() {
        delete[] data_;
        data_ = nullptr;
    }

    bool      empty()    const noexcept { return size_ == 0; }
    size_type size()     const noexcept { return size_; }
    size_type max_size() const noexcept { return 1'010'000'000; }

    iterator       begin()        noexcept { return data_; }
    const_iterator begin()  const noexcept { return data_; }
    const_iterator cbegin() const noexcept { return data_; }

    iterator       end()        noexcept { return data_+size_; }
    const_iterator end()  const noexcept { return data_+size_; }
    const_iterator cend() const noexcept { return data_+size_; }

    reverse_iterator       rbegin()        noexcept { return reverse_iterator(end()); }
    const_reverse_iterator rbegin()  const noexcept { return const_reverse_iterator(end()); }
    const_reverse_iterator crbegin() const noexcept { return const_reverse_iterator(end()); }

    reverse_iterator       rend()        noexcept { return reverse_iterator(begin()); }
    const_reverse_iterator rend()  const noexcept { return const_reverse_iterator(begin()); }
    const_reverse_iterator crend() const noexcept { return const_reverse_iterator(begin()); }

    reference       operator[](size_type pos)       { return data_[pos]; }
    const_reference operator[](size_type pos) const { return data_[pos]; }

    bool*       data()       noexcept { return data_; }
    const bool* data() const noexcept { return data_; }

private:
    size_t size_;
    bool*  data_;
};

void swap(BoolArray& lhs, BoolArray& rhs) noexcept { lhs.swap(rhs); }

bool operator==(const BoolArray& lhs, const BoolArray& rhs) {
    return equal(begin(lhs), end(lhs), begin(rhs), end(rhs));
}
bool operator!=(const BoolArray& lhs, const BoolArray& rhs) { return !(lhs == rhs); }

bool operator<(const BoolArray& lhs, const BoolArray& rhs) {
    return lexicographical_compare(begin(lhs), end(lhs), begin(rhs), end(rhs));
}
bool operator> (const BoolArray& lhs, const BoolArray& rhs) { return rhs < lhs; }
bool operator<=(const BoolArray& lhs, const BoolArray& rhs) { return !(rhs < lhs); }
bool operator>=(const BoolArray& lhs, const BoolArray& rhs) { return !(lhs < rhs); }
// }}}

template<typename K, typename V, typename Comp>
bool map_contains(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename V, typename Hash, typename Eq>
bool map_contains(const unordered_map<K,V,Hash,Eq>& m, const typename unordered_map<K,V,Hash,Eq>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename Comp>
bool multiset_erase_one(multiset<K,Comp>& m, const typename multiset<K,Comp>::key_type& k) {
    auto it = m.find(k);
    if(it == end(m)) return false;
    m.erase(it);
    return true;
}

template<typename K, typename Hash, typename Eq>
bool multiset_erase_one(unordered_multiset<K,Hash,Eq>& m, const typename unordered_multiset<K,Hash,Eq>::key_type& k) {
    auto it = m.find(k);
    if(it == end(m)) return false;
    m.erase(it);
    return true;
}

template<typename T>
using MaxHeap = priority_queue<T, vector<T>, less<T>>;
template<typename T>
using MinHeap = priority_queue<T, vector<T>, greater<T>>;

// {{{ POP() 系
// 効率は悪い
template<typename T>
T POP_FRONT(vector<T>& v) {
    T x = v.front(); v.erase(begin(v));
    return x;
}

template<typename T>
T POP_BACK(vector<T>& v) {
    T x = v.back(); v.pop_back();
    return x;
}

template<typename T>
T POP_FRONT(deque<T>& v) {
    T x = v.front(); v.pop_front();
    return x;
}

template<typename T>
T POP_BACK(deque<T>& v) {
    T x = v.back(); v.pop_back();
    return x;
}

template<typename T>
T POP_FRONT(forward_list<T>& ls) {
    T x = ls.front(); ls.pop_front();
    return x;
}

template<typename T>
T POP_FRONT(list<T>& ls) {
    T x = ls.front(); ls.pop_front();
    return x;
}

template<typename T>
T POP_BACK(list<T>& ls) {
    T x = ls.back(); ls.pop_back();
    return x;
}

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;
}
// }}}

// {{{ Formatter
template<typename T, size_t N>
struct Formatter<array<T,N>> {
    static ostream& write_str(ostream& out, const array<T,N>& a) {
        return WRITE_RANGE_STR(out, begin(a), end(a));
    }
    static ostream& write_repr(ostream& out, const array<T,N>& a) {
        out << "array";
        return WRITE_RANGE_REPR(out, begin(a), end(a));
    }
};

template<typename T>
struct Formatter<deque<T>> {
    static ostream& write_str(ostream& out, const deque<T>& deq) {
        return WRITE_RANGE_STR(out, begin(deq), end(deq));
    }
    static ostream& write_repr(ostream& out, const deque<T>& deq) {
        out << "deque";
        return WRITE_RANGE_REPR(out, begin(deq), end(deq));
    }
};

template<typename T>
struct Formatter<forward_list<T>> {
    static ostream& write_str(ostream& out, const forward_list<T>& ls) {
        return WRITE_RANGE_STR(out, begin(ls), end(ls));
    }
    static ostream& write_repr(ostream& out, const forward_list<T>& ls) {
        out << "forward_list";
        return WRITE_RANGE_REPR(out, begin(ls), end(ls));
    }
};

template<typename T>
struct Formatter<list<T>> {
    static ostream& write_str(ostream& out, const list<T>& ls) {
        return WRITE_RANGE_STR(out, begin(ls), end(ls));
    }
    static ostream& write_repr(ostream& out, const list<T>& ls) {
        out << "list";
        return WRITE_RANGE_REPR(out, begin(ls), end(ls));
    }
};

template<typename T, typename Comp>
struct Formatter<set<T,Comp>> {
    static ostream& write_str(ostream& out, const set<T,Comp>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const set<T,Comp>& s) {
        out << "set";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Comp>
struct Formatter<multiset<T,Comp>> {
    static ostream& write_str(ostream& out, const multiset<T,Comp>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const multiset<T,Comp>& s) {
        out << "multiset";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Hash, typename Eq>
struct Formatter<unordered_set<T,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_set<T,Hash,Eq>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const unordered_set<T,Hash,Eq>& s) {
        out << "unordered_set";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Hash, typename Eq>
struct Formatter<unordered_multiset<T,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_multiset<T,Hash,Eq>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const unordered_multiset<T,Hash,Eq>& s) {
        out << "unordered_multiset";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename K, typename V, typename Comp>
struct Formatter<map<K,V,Comp>> {
    static ostream& write_str(ostream& out, const map<K,V,Comp>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const map<K,V,Comp>& m) {
        out << "map";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Comp>
struct Formatter<multimap<K,V,Comp>> {
    static ostream& write_str(ostream& out, const multimap<K,V,Comp>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const multimap<K,V,Comp>& m) {
        out << "multimap";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Hash, typename Eq>
struct Formatter<unordered_map<K,V,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_map<K,V,Hash,Eq>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const unordered_map<K,V,Hash,Eq>& m) {
        out << "unordered_map";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Hash, typename Eq>
struct Formatter<unordered_multimap<K,V,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_multimap<K,V,Hash,Eq>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const unordered_multimap<K,V,Hash,Eq>& m) {
        out << "unordered_multimap";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename T, typename C>
struct Formatter<stack<T,C>> {
    static ostream& write_str(ostream& out, const stack<T,C>& orig) {
        stack<T,C> stk(orig);
        while(!stk.empty()) {
            WRITE_STR(out, stk.top()); stk.pop();
            if(!stk.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const stack<T,C>& orig) {
        stack<T,C> stk(orig);
        out << "stack[";
        while(!stk.empty()) {
            WRITE_REPR(out, stk.top()); stk.pop();
            if(!stk.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};

template<typename T, typename C>
struct Formatter<queue<T,C>> {
    static ostream& write_str(ostream& out, const queue<T,C>& orig) {
        queue<T,C> que(orig);
        while(!que.empty()) {
            WRITE_STR(out, que.front()); que.pop();
            if(!que.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const queue<T,C>& orig) {
        queue<T,C> que(orig);
        out << "queue[";
        while(!que.empty()) {
            WRITE_REPR(out, que.front()); que.pop();
            if(!que.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};

template<typename T, typename C, typename Comp>
struct Formatter<priority_queue<T,C,Comp>> {
    static ostream& write_str(ostream& out, const priority_queue<T,C,Comp>& orig) {
        priority_queue<T,C,Comp> que(orig);
        while(!que.empty()) {
            WRITE_STR(out, que.top()); que.pop();
            if(!que.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const priority_queue<T,C,Comp>& orig) {
        priority_queue<T,C,Comp> que(orig);
        out << "priority_queue[";
        while(!que.empty()) {
            WRITE_REPR(out, que.top()); que.pop();
            if(!que.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};

template<>
struct Formatter<BoolArray> {
    static ostream& write_str(ostream& out, const BoolArray& a) {
        return WRITE_RANGE_STR(out, begin(a), end(a));
    }
    static ostream& write_repr(ostream& out, const BoolArray& a) {
        out << "BoolArray";
        return WRITE_RANGE_REPR(out, begin(a), end(a));
    }
};
// }}}

// }}}

// {{{ graph (container が必要)

// ダイクストラ法
//
// (d,parent) を返す
// d[i]: start から点 i への最短距離(到達不能な点は INF)
// parent[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
pair<vector<i64>,vector<i64>> graph_dijkstra(const vector<vector<pair<i64,i64>>>& g, i64 start) {
    i64 n = SIZE(g);
    vector<i64> d(n, INF);
    vector<i64> parent(n, -1);

    BoolArray done(n, false);
    MinHeap<pair<i64,i64>> que;

    d[start] = 0;
    que.emplace(0, start);

    i64 n_remain = n;
    while(!que.empty()) {
        i64 dmin,vmin; tie(dmin,vmin) = POP(que);

        if(d[vmin] < dmin) continue;

        done[vmin] = true;
        if(--n_remain == 0) break;

        for(const auto& p : g[vmin]) {
            i64 to,cost; tie(to,cost) = p;
            if(done[to]) continue;

            i64 d_new = dmin + cost;
            if(d_new < d[to]) {
                d[to] = d_new;
                parent[to] = vmin;
                que.emplace(d_new, to);
            }
        }
    }

    return { d, parent };
}

// ベルマンフォード法
//
// (ok,d,parent) を返す
// ok: 負閉路が存在しない場合に限り true
// d[i]: start から点 i への最短距離(到達不能な点は INF, 負閉路上の点は -INF)
// parent[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
tuple<bool,vector<i64>,vector<i64>> graph_bellman(const vector<vector<pair<i64,i64>>>& g, i64 start) {
    i64 n = SIZE(g);
    bool ok = true;
    vector<i64> d(n, INF);
    vector<i64> parent(n, -1);

    d[start] = 0;

    REP(i, n) {
        bool update = false;
        REP(from, n) {
            if(d[from] == INF) continue;
            for(const auto& p : g[from]) {
                i64 to,cost; tie(to,cost) = p;
                i64 d_new = d[from] + cost;
                if(d_new < d[to]) {
                    update = true;
                    d[to] = i == n-1 ? -INF : d_new;
                    parent[to] = from;
                }
            }
        }
        if(!update) break;
        if(i == n-1) ok = false;
    }

    return make_tuple(ok, d, parent);
}

// SPFA (Shortest Path Faster Algorithm)
//
// ベルマンフォードより速い、はず
//
// (ok,d,parent) を返す
// ok: 負閉路が存在しない場合に限り true
// d[i]: start から点 i への最短距離(到達不能な点は INF, 負閉路上の点は -INF)
// parent[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
tuple<bool,vector<i64>,vector<i64>> graph_spfa(const vector<vector<pair<i64,i64>>>& g, i64 start) {
    i64 n = SIZE(g);
    bool ok = true;
    vector<i64> d(n, INF);
    vector<i64> parent(n, -1);

    queue<i64> que;
    BoolArray in_que(n, false);
    const auto enqueue = [&que,&in_que](i64 v) { que.emplace(v); in_que[v] = true; };
    const auto dequeue = [&que,&in_que]() { i64 v = POP(que); in_que[v] = false; return v; };

    d[start] = 0;
    enqueue(start);

    REP(i, n) {
        REP(_, que.size()) {
            i64 from = dequeue();
            for(const auto& p : g[from]) {
                i64 to,cost; tie(to,cost) = p;
                i64 d_new = d[from] + cost;
                if(d_new < d[to]) {
                    d[to] = i == n-1 ? -INF : d_new;
                    parent[to] = from;
                    if(!in_que[to]) enqueue(to);
                }
            }
        }
        if(que.empty()) break;
        if(i == n-1) ok = false;
    }

    return make_tuple(ok, d, parent);
}

// ワーシャルフロイド法
//
// g は隣接行列 (g[from][to]) で、from == to の場合 0, from != to で辺
// がない場合 INF
//
// g は全点対間最短距離で上書きされる
// (ok,nex) を返す
// ok: 負閉路が存在しない場合に限り true
// nex[i][j]: i から j へ最短経路で行くとき、次に辿るべき点
pair<bool,vector<vector<i64>>> graph_floyd(vector<vector<f64>>& g) {
    i64 n = SIZE(g);
    vector<vector<i64>> nex(n, vector<i64>(n,-1));
    REP(i, n) REP(j, n) {
        if(g[i][j] != 1e100)
            nex[i][j] = j;
    }

    REP(k, n) {
        REP(i, n) {
            if(g[i][k] == 1e100) continue;
            REP(j, n) {
                if(g[k][j] == 1e100) continue;
                if(chmin(g[i][j], g[i][k] + g[k][j])) {
                    nex[i][j] = nex[i][k];
                }
                if(i == j && g[i][j] < 0) return { false, nex };
            }
        }
    }

    return { true, nex };
}

// }}}

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

constexpr i64 ipow(i64 x, i64 e) {
    assert(e >= 0);
    i64 res = 1;
    REP(_, e) {
        res *= x;
    }
    return res;
}

struct Pos {
    i64 x, y;
};

void solve(i64 N) {
    vector<Pos> ps(N);
    REP(_, N) {
        i64 b; RD(b); --b;  // 0-based
        i64 x,y; RD(x); RD(y);
        ps[b] = { x, y };
    }

    vector<vector<f64>> G(N, vector<f64>(N, 1e100));
    REP(i, N) REP(j, N) {
        const Pos& p1 = ps[i];
        const Pos& p2 = ps[j];
        i64 d_sq = ipow(p1.x-p2.x,2) + ipow(p1.y-p2.y,2);
        if(d_sq <= 2500) {
            G[i][j] = G[j][i] = sqrt(d_sq);
        }
    }

    bool ok;
    vector<vector<i64>> nex;
    tie(ok,nex) = graph_floyd(G);
    assert(ok);

    i64 M; RD(M);
    REP(_, M) {
        i64 s,g; RD(s); RD(g); --s; --g;  // 0-based
        if(nex[s][g] == -1) {
            PRINTLN("NA");
            continue;
        }

        vector<i64> route;
        for(i64 v = s; v != g; v = nex[v][g]) {
            route.emplace_back(v);
        }
        route.emplace_back(g);

        ALL(transform_self, route, [](i64 e) { return e+1; });  // 1-based
        PRINTLN(route);
    }

    // * MOD はとった?
    // * 違うやつ提出してない?
    // * 違うやつテストしてない?
    //PRINTLN(ans);
}

signed main() {
    for(;;) {
        i64 N; RD(N);
        if(N == 0) break;
        solve(N);
    }

    EXIT();
}



Measured memory_bytes: 3172