#include <bits/stdc++.h>
// created [2019/12/03] 23:59:35
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"

using i32 = int32_t;
using i64 = int64_t;
using u32 = uint32_t;
using u64 = uint64_t;
using uint = unsigned int;
using usize = std::size_t;
using ll = long long;
using ull = unsigned long long;
using ld = long double;
template<typename T> constexpr T popcount(const T u) { return u ? static_cast<T>(__builtin_popcountll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T log2p1(const T u) { return u ? static_cast<T>(64 - __builtin_clzll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T msbp1(const T u) { return log2p1(u); }
template<typename T> constexpr T lsbp1(const T u) { return __builtin_ffsll(u); }
template<typename T> constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast<T>(u); }
template<typename T> constexpr bool ispow2(const T u) { return u and (static_cast<u64>(u) & static_cast<u64>(u - 1)) == 0; }
template<typename T> constexpr T ceil2(const T u) { return static_cast<T>(1) << clog(u); }
template<typename T> constexpr T floor2(const T u) { return u == 0 ? static_cast<T>(0) : static_cast<T>(1) << (log2p1(u) - 1); }
template<typename T> constexpr bool btest(const T mask, const usize ind) { return static_cast<bool>((static_cast<u64>(mask) >> ind) & static_cast<u64>(1)); }
template<typename T> void bset(T& mask, const usize ind) { mask |= (static_cast<T>(1) << ind); }
template<typename T> void breset(T& mask, const usize ind) { mask &= ~(static_cast<T>(1) << ind); }
template<typename T> void bflip(T& mask, const usize ind) { mask ^= (static_cast<T>(1) << ind); }
template<typename T> void bset(T& mask, const usize ind, const bool b) { (b ? bset(mask, ind) : breset(mask, ind)); }
template<typename T> constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast<T>(0) : static_cast<T>((static_cast<u64>(mask) << (64 - ind)) >> (64 - ind)); }
template<typename T> bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); }
template<typename T> bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); }
constexpr unsigned int mod                  = 1000000007;
template<typename T> constexpr T inf_v      = std::numeric_limits<T>::max() / 4;
template<typename Real> constexpr Real pi_v = Real{3.141592653589793238462643383279502884};

template<typename T>
T read()
{
    T v;
    return std::cin >> v, v;
}
template<typename T, typename... Args>
auto read(const usize size, Args... args)
{
    std::vector<decltype(read<T>(args...))> ans(size);
    for (usize i = 0; i < size; i++) { ans[i] = read<T>(args...); }
    return ans;
}
template<typename... Types>
auto reads() { return std::tuple<std::decay_t<Types>...>{read<Types>()...}; }
#    define SHOW(...) static_cast<void>(0)
constexpr ull TEN(const usize n) { return n == 0 ? 1ULL : TEN(n - 1) * 10ULL; }

template<typename T>
std::vector<T> make_v(const usize size, const T v) { return std::vector<T>(size, v); }
template<typename... Args>
auto make_v(const usize size, Args... args) { return std::vector<decltype(make_v(args...))>(size, make_v(args...)); }


template<typename T> T gcd(const T& a, const T& b) { return a < 0 ? gcd(-a, b) : b < 0 ? gcd(a, -b) : (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); }
template<typename T> T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; }
template<typename T>
constexpr std::pair<T, T> extgcd(const T a, const T b)
{
    if (b == 0) { return std::pair<T, T>{1, 0}; }
    const auto g = gcd(a, b), da = std::abs(b) / g;
    const auto p = extgcd(b, a % b);
    const auto x = (da + p.second % da) % da, y = (g - a * x) / b;
    return {x, y};
}
template<typename T>
constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; }
template<uint mod_value, bool dynamic = false>
class modint_base
{
public:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, const UInt> mod() { return mod_ref(); }
    template<typename UInt = uint>
    static constexpr std::enable_if_t<not dynamic, const UInt> mod() { return mod_value; }
    template<typename UInt = uint>
    static void set_mod(const std::enable_if_t<dynamic, const UInt> mod) { mod_ref() = mod, inv_ref() = {1, 1}; }
    modint_base() : v{0} {}
    modint_base(const ll val) : v{norm(static_cast<uint>(val % static_cast<ll>(mod()) + static_cast<ll>(mod())))} {}
    modint_base(const modint_base& n) : v{n()} {}
    explicit operator bool() const { return v != 0; }
    bool operator!() const { return not static_cast<bool>(*this); }
    modint_base& operator=(const modint_base& m) { return v = m(), (*this); }
    modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast<ll>(mod()) + static_cast<ll>(mod()))), (*this); }
    friend modint_base operator+(const modint_base& m) { return m; }
    friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); }
    friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); }
    friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); }
    friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast<uint>(static_cast<ll>(m1.v) * static_cast<ll>(m2.v) % static_cast<ll>(mod()))); }
    friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); }
    friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) - val}; }
    friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * inv(val)}; }
    friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast<ll>(m.v) + val}; }
    friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast<ll>(m.v))}; }
    friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; }
    friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; }
    friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; }
    friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; }
    friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; }
    friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; }
    friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; }
    friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; }
    friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); }
    friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; }
    friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; }
    friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); }
    friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); }
    friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); }
    friend std::istream& operator>>(std::istream& is, modint_base& m)
    {
        ll v;
        return is >> v, m = v, is;
    }
    friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); }
    uint operator()() const { return v; }
    static modint_base small_inv(const usize n)
    {
        auto& in = inv_ref();
        if (n < in.size()) { return in[n]; }
        for (usize i = in.size(); i <= n; i++) { in.push_back(-in[modint_base::mod() % i] * (modint_base::mod() / i)); }
        return in.back();
    }

private:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, UInt&> mod_ref()
    {
        static UInt mod = 0;
        return mod;
    }
    static uint norm(const uint x) { return x < mod() ? x : x - mod(); }
    static modint_base make(const uint x)
    {
        modint_base m;
        return m.v = x, m;
    }
    static modint_base power(modint_base x, ull n)
    {
        modint_base ans = 1;
        for (; n; n >>= 1, x *= x) {
            if (n & 1) { ans *= x; }
        }
        return ans;
    }
    static modint_base inv(const ll v) { return v < 1000000 ? small_inv(static_cast<usize>(v)) : modint_base{inverse(v, static_cast<ll>(mod()))}; }
    static std::vector<modint_base>& inv_ref()
    {
        static std::vector<modint_base> in{1, 1};
        return in;
    }
    uint v;
};
template<uint mod>
using modint = modint_base<mod, false>;
template<uint id>
using dynamic_modint = modint_base<id, true>;


/**
 * http://xoshiro.di.unimi.it/xoshiro128starstar.c
 * http://xoshiro.di.unimi.it/xoshiro256starstar.c
 * http://xoshiro.di.unimi.it/splitmix64.c
 */
class xoshiro
{
public:
    using result_type = uint32_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro() : xoshiro(std::random_device{}()) {}
    xoshiro(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 33)) * 0x62A9D9ED799705F5, z = (z ^ (z >> 28)) * 0xCB24D0A5C88C35B3, s[i] = static_cast<result_type>(z >> 32); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 9;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 11), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (32 - k)); }
};
class xoshiro_64
{
public:
    using result_type = uint64_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro_64() : xoshiro_64(std::random_device{}()) {}
    xoshiro_64(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9, z = (z ^ (z >> 27)) * 0x94d049bb133111eb, s[i] = static_cast<result_type>(z ^ (z >> 31)); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 17;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 45), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (64 - k)); }
};
template<typename Rng>
class rng_base
{
public:
    using rng_type    = Rng;
    using result_type = typename rng_type::result_type;
    static constexpr result_type min() { return rng_type::min(); }
    static constexpr result_type max() { return rng_type::max(); }
    rng_base() : rng_base(std::random_device{}()) {}
    rng_base(const u64 seed) : rng(seed) {}
    ~rng_base() = default;
    result_type operator()(const result_type max = std::numeric_limits<result_type>::max())
    {
        if (max == std::numeric_limits<result_type>::max()) { return static_cast<result_type>(rng()); }
        if (ispow2(max + 1)) { return static_cast<result_type>(rng() & max); }
        const result_type mask = static_cast<result_type>(ceil2(static_cast<u64>(max + 1))) - 1;
        while (true) {
            const result_type ans = static_cast<result_type>(rng() & mask);
            if (ans <= max) { return ans; }
        }
    }
    result_type operator()(const result_type min, const result_type max) { return min + (*this)(max - min); }
    operator bool() { return *this(0, 1); }
    template<typename Int> std::pair<Int, Int> pair(const Int min, const Int max, const bool sorted = false) { return sorted ? std::minmax(*this(min, max), *this(min, max)) : std::pair<Int, Int>{*this(min, max), *this(min, max)}; }
    template<typename Int>
    std::vector<Int> vec(const std::size_t size, const Int min, const Int max)
    {
        std::vector<Int> v(size);
        for (std::size_t i = 0; i < size; i++) { v[i] = *this(min, max); }
        return v;
    }
    std::vector<usize> perm(const usize n)
    {
        std::vector<usize> ans(n);
        std::iota(ans.begin(), ans.end(), 0UL);
        std::shuffle(ans.begin(), ans.end(), rng);
        return ans;
    }

private:
    Rng rng;
};
using rng_mt        = rng_base<std::mt19937>;
using rng_mt64      = rng_base<std::mt19937_64>;
using rng_xoshiro   = rng_base<xoshiro>;
using rng_xoshiro64 = rng_base<xoshiro_64>;
rng_mt g_rng_mt;
rng_mt64 g_rng_mt64;
rng_xoshiro g_rng_xo;
rng_xoshiro64 g_rng_xo64;


namespace bbst_node {
template<typename Key, typename Node, typename Comp>
struct key_node : Node
{
    using ptr       = key_node* const;
    using const_ptr = const ptr;
    using key_type  = Key;
    using comp_type = Comp;
    key_node() : Node{}, key{Key{}} {}
    template<typename... Args>
    key_node(const Key& key, Args... args) : Node{args...}, key{key} { this->sz = 1; }
    void pull_up(const_ptr l, const_ptr r) { Node::pull_up(l, r); }
    void push_down(ptr l, ptr r) { Node::push_down(l, r); }
    template<typename Value>
    void set(const Value& val, const_ptr l, const_ptr r) { Node::set(val, l, r); }
    template<typename Op>
    void act(const Op& o) { Node::act(o); }
    friend std::ostream& operator<<(std::ostream& os, const key_node& n) { return os << "key=" << n.key << ":" << static_cast<Node>(n); }
    const key_type key;
};
struct node
{
    using ptr       = node* const;
    using const_ptr = const ptr;
    void pull_up(const_ptr l, const_ptr r) { sz = (l ? l->sz : 0UL) + 1UL + (r ? r->sz : 0UL); }
    void push_down(ptr, ptr) {}
    friend std::ostream& operator<<(std::ostream& os, const node& n) { return os << "size=" << n.sz; }
    usize sz = 0;
};
template<typename Value>
struct value_node
{
    using ptr        = value_node* const;
    using const_ptr  = const ptr;
    using value_type = Value;
    value_node()     = default;
    value_node(const value_type& value) : value{value}, sz{1} {}
    void pull_up(const_ptr l, const_ptr r) { sz = (l ? l->sz : 0UL) + 1UL + (r ? r->sz : 0UL); }
    void push_down(ptr, ptr) {}
    void set(const value_type& val, const_ptr, const_ptr) { value = val; }
    friend std::ostream& operator<<(std::ostream& os, const value_node& n) { return os << "value=" << n.value; }
    value_type value{};
    usize sz = 0;
};
template<typename ValueMonoid>
struct merge_node
{
    using ptr               = merge_node* const;
    using const_ptr         = const ptr;
    using value_monoid_type = ValueMonoid;
    using value_type        = typename value_monoid_type::value_type;
    merge_node()            = default;
    merge_node(const value_type& value) : value{value}, merged{value}, sz{1} {}
    void pull_up(const_ptr l, const_ptr r) { sz = (l ? l->sz : 0UL) + 1UL + (r ? r->sz : 0UL), merged = value_monoid_type::merge((l ? l->merged : value_monoid_type::id()), value_monoid_type::merge(value, (r ? r->merged : value_monoid_type::id()))); }
    void push_down(ptr, ptr) {}
    void set(const value_type& val, const_ptr l, const_ptr r) { value = val, pull_up(l, r); }
    friend std::ostream& operator<<(std::ostream& os, const merge_node& n) { return os << "value=" << n.value << ",merged=" << n.merged; }
    value_type value = value_monoid_type::id(), merged = value_monoid_type::id();
    usize sz = 0;
};
template<typename MonoidAct>
struct lazy_node
{
    using ptr                  = lazy_node* const;
    using const_ptr            = const ptr;
    using monoid_act_type      = MonoidAct;
    using value_monoid_type    = typename monoid_act_type::value_monoid_type;
    using operator_monoid_type = typename monoid_act_type::operator_monoid_type;
    using value_type           = typename value_monoid_type::value_type;
    using operator_type        = typename operator_monoid_type::operator_type;
    lazy_node()                = default;
    lazy_node(const value_type& value) : value{value}, merged{value}, sz{1} {}
    void pull_up(const_ptr l, const_ptr r) { sz = (l ? l->sz : 0UL) + 1UL + (r ? r->sz : 0UL), merged = value_monoid_type::merge((l ? l->merged : value_monoid_type::id()), value_monoid_type::merge(value, (r ? r->merged : value_monoid_type::id()))); }
    void push_down(ptr l, ptr r)
    {
        if (op == operator_monoid_type::id()) { return; }
        if (l) { l->act(op); }
        if (r) { r->act(op); }
        op = operator_monoid_type::id();
    }
    void set(const value_type& val, const_ptr l, const_ptr r) { value = val, pull_up(l, r); }
    void act(const operator_type& o) { value = monoid_act_type::apply(o, value, 1), merged = monoid_act_type::apply(o, merged, sz), op = operator_monoid_type::compose(op, o); }
    friend std::ostream& operator<<(std::ostream& os, const lazy_node& n) { return os << "value=" << n.value << ",merged=" << n.merged << ",op=" << n.op; }
    value_type value = value_monoid_type::id(), merged = value_monoid_type::id();
    operator_type op = operator_monoid_type::id();
    usize sz         = 0;
};
}  // namespace bbst_node
template<typename NodeData>
class base_rbstree
{
private:
    class node
    {
        using ptr       = node*;
        using const_ptr = const node* const;
        using tree      = ptr;

    public:
        node() : node_data{} {}
        template<typename... Args>
        node(Args... args) : node_data{args...} {}
        usize size() const { return node_data.sz; }
        const NodeData& data() const { return node_data; }
        template<typename Operator>
        void act(const Operator& op) { node_data.act(op); }
        template<typename Value>
        void set(const Value& value) { node_data.set(value, data_ptr_of(l), data_ptr_of(r)), pull_up(); }
        friend std::ostream& operator<<(std::ostream& os, const node& n) { return os << "[" << n.node_data << "]"; }
        bool has_left() const { return l; }
        bool has_right() const { return r; }
        const node& left() const { return deptr(l); }
        const node& right() const { return deptr(r); }
        static node& deptr(ptr x) { return x ? (*x) : empty_node; }
        static ptr merge(tree tp1, tree tp2)
        {
            if (not tp1) { return tp2; }
            if (not tp2) { return tp1; }
            if (g_rng_xo(static_cast<uint>(size_of(tp1) + size_of(tp2) - 1)) < static_cast<uint>(size_of(tp1))) {
                return tp1->push_down(), tp1->r = merge(tp1->r, tp2), tp1->pull_up(), tp1;
            } else {
                return tp2->push_down(), tp2->l = merge(tp1, tp2->l), tp2->pull_up(), tp2;
            }
        }
        static std::pair<ptr, ptr> split_at(tree tp, const usize pos)
        {
            if (not tp) { return {nullptr, nullptr}; }
            tp->push_down();
            if (pos == 0) { return std::make_pair(nullptr, tp); }
            if (pos == size_of(tp)) { return std::make_pair(tp, nullptr); }
            if (pos <= size_of(tp->l)) {
                auto ls      = split_at(tp->l, pos);
                return tp->l = ls.second, tp->pull_up(), std::make_pair(ls.first, tp);
            } else {
                auto rs      = split_at(tp->r, pos - size_of(tp->l) - 1);
                return tp->r = rs.first, tp->pull_up(), std::make_pair(tp, rs.second);
            }
        }
        static std::tuple<ptr, ptr, ptr> split_range(tree tp, const usize pos_min, const usize pos_sup)
        {
            auto ts = split_at(tp, pos_min), trs = split_at(ts.second, pos_sup - pos_min);
            return std::make_tuple(ts.first, trs.first, trs.second);
        }
        template<typename Key>
        static std::pair<ptr, ptr> split_lower(tree tp, const Key& key)
        {
            if (not tp) { return {nullptr, nullptr}; }
            const typename NodeData::comp_type& comp{};
            if (comp(tp->data().key, key)) {
                auto rs      = split_lower(tp->r, key);
                return tp->r = rs.first, tp->pull_up(), std::make_pair(tp, rs.second);
            } else {
                auto ls      = split_lower(tp->l, key);
                return tp->l = ls.second, tp->pull_up(), std::make_pair(ls.first, tp);
            }
        }
        template<typename Key>
        static std::pair<ptr, ptr> split_upper(tree tp, const Key& key)
        {
            if (not tp) { return {nullptr, nullptr}; }
            const typename NodeData::comp_type& comp{};
            if (comp(key, tp->data().key)) {
                auto ls      = split_upper(tp->l, key);
                return tp->l = ls.second, tp->pull_up(), std::make_pair(ls.first, tp);
            } else {
                auto rs      = split_upper(tp->r, key);
                return tp->r = rs.first, tp->pull_up(), std::make_pair(tp, rs.second);
            }
        }
        template<typename Key>
        static std::tuple<ptr, ptr, ptr> split_key_range(tree tp, const Key& key_min, const Key& key_max)
        {
            auto ts = split_lower(tp, key_min), trs = split_upper(ts.second, key_max);
            return std::make_tuple(ts.first, trs.first, trs.second);
        }

    private:
        static NodeData* data_ptr_of(ptr x) { return x ? &(x->node_data) : nullptr; }
        static usize size_of(const_ptr x) { return x ? x->size() : 0UL; }
        void pull_up() { node_data.pull_up(data_ptr_of(l), data_ptr_of(r)); }
        void push_down() { node_data.push_down(data_ptr_of(l), data_ptr_of(r)); }
        static node empty_node;  // inline変数を使いたいでござる
        NodeData node_data;
        ptr l = nullptr, r = nullptr;
    };
    using ptr = node*;
    ptr root  = nullptr;

public:
    base_rbstree() = default;
    base_rbstree(const ptr r) : root{r} {}
    template<typename... Args>
    base_rbstree(Args... args) : root{new node{args...}} {}
    bool empty() const { return not root; }
    usize size() const { return node::size(root); }
    const node& top() const { return node::deptr(root); }
    node at(const usize pos) { return fold_range(pos, pos + 1); }
    node fold_range(const usize pos_min, const usize pos_sup)
    {
        auto ts        = node::split_range(root, pos_min, pos_sup);
        const node ans = node::deptr(std::get<1>(ts));
        return root    = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), ans;
    }
    template<typename Key>
    node fold_key_range(const Key& key_min, const Key& key_max)
    {
        auto ts        = node::split_key_range(root, key_min, key_max);
        const node ans = node::deptr(std::get<1>(ts));
        return root    = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), ans;
    }
    template<typename Key>
    node fold_lower(const Key& key)
    {
        auto ts        = node::split_lower(root, key);
        const node ans = node::deptr(ts.first);
        return root    = node::merge(ts.first, ts.second), ans;
    }
    template<typename Key>
    node fold_upper(const Key& key)
    {
        auto ts        = node::split_upper(root, key);
        const node ans = node::deptr(ts.first);
        return root    = node::merge(ts.first, ts.second), ans;
    }
    template<typename Value>
    base_rbstree& set_at(const usize pos, const Value& value)
    {
        auto ts = node::split_range(root, pos, pos + 1);
        std::get<1>(ts)->set(value);
        return root = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), *this;
    }
    template<typename Key, typename Value>
    base_rbstree& set(const Key& key, const Value& value)
    {
        auto ts = node::split_key_range(root, key, key);
        std::get<1>(ts)->set(value);
        return root = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), *this;
    }
    template<typename Op>
    base_rbstree& act_range(const usize pos_min, const usize pos_sup, const Op& op)
    {
        auto ts = node::split_range(root, pos_min, pos_sup);
        std::get<1>(ts)->act(op);
        return root = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), *this;
    }
    template<typename Key, typename Op>
    base_rbstree& act_key_range(const Key& key_min, const Key& key_max, const Op& op)
    {
        auto ts = node::split_key_range(root, key_min, key_max);
        std::get<1>(ts)->act(op);
        return root = node::merge(std::get<0>(ts), node::merge(std::get<1>(ts), std::get<2>(ts))), *this;
    }
    base_rbstree& merge(base_rbstree&& t) { return root = node::merge(root, t.root), *this; }
    base_rbstree split_at(const usize pos)
    {
        auto ts     = node::split_at(root, pos);
        return root = ts.first, base_rbstree(ts.second);
    }
    std::pair<base_rbstree, base_rbstree> split_range(const usize pos_min, const usize pos_sup)
    {
        auto ts     = node::split_range(root, pos_min, pos_sup);
        return root = std::get<0>(ts), std::make_pair(base_rbstree(std::get<1>(ts)), base_rbstree(std::get<2>(ts)));
    }
    base_rbstree& erase_at(const usize pos)
    {
        auto ts = node::split_at(root, pos), trs = node::split_at(ts.second, 1);
        return root = node::merge(ts.first, trs.second), *this;
    }
    base_rbstree& insert_at(const usize pos, base_rbstree&& t)
    {
        auto ts     = node::split_at(root, pos);
        return root = node::merge(node::merge(ts.first, t.root), ts.second), *this;
    }
    template<typename Key>
    base_rbstree split_lower(const Key& key)
    {
        auto ts     = node::split_lower(root, key);
        return root = ts.first, base_rbstree(ts.second);
    }
    template<typename Key>
    base_rbstree split_upper(const Key& key)
    {
        auto ts     = node::split_upper(root, key);
        return root = ts.first, base_rbstree(ts.second);
    }
    template<typename Key>
    std::pair<base_rbstree, base_rbstree> split_key_range(const Key& key_min, const Key& key_max)
    {
        auto ts     = node::split_key_range(root, key_min, key_max);
        return root = std::get<0>(ts), std::make_pair(base_rbstree(std::get<1>(ts)), base_rbstree(std::get<2>(ts)));
    }
    base_rbstree& insert(const node& n)
    {
        auto ts     = node::split_lower(root, n.data().key);
        return root = node::merge(node::merge(ts.first, new node{n}), ts.second), *this;
    }
    std::vector<node> data()
    {
        if (empty()) { return std::vector<node>{}; }
        auto dfs = [&](auto&& self, const node& n) -> std::vector<node> {
            std::vector<node> ans;
            if (n.has_left()) {
                for (auto&& e : self(self, n.left())) { ans.emplace_back(e); }
            }
            ans.push_back(n);
            if (n.has_right()) {
                for (auto&& e : self(self, n.right())) { ans.emplace_back(e); }
            }
            return ans;
        };
        return dfs(dfs, *root);
    }
};
template<typename NodeData>
typename base_rbstree<NodeData>::node base_rbstree<NodeData>::node::empty_node = node{};
template<typename Value>
using rbstree = base_rbstree<bbst_node::value_node<Value>>;
template<typename Key, typename Comp = std::less<Key>>
using mset_rbstree = base_rbstree<bbst_node::key_node<Key, bbst_node::node, Comp>>;
template<typename Key, typename Value, typename Comp = std::less<Key>>
using mmap_rbstree = base_rbstree<bbst_node::key_node<Key, bbst_node::value_node<Value>, Comp>>;
template<typename ValueMonoid>
using merge_rbstree = base_rbstree<bbst_node::merge_node<ValueMonoid>>;
template<typename MonoidAct>
using lazy_rbstree = base_rbstree<bbst_node::lazy_node<MonoidAct>>;
int main()
{
    const int N = read<int>();
    const ull X = read<ull>();
    auto A      = read<ull>(N);

    {
        ull x = X + 1;
        for (const ull a : A) { x /= (a + 1); }
        if (x == 0) { return std::cout << std::string(X, '0') << std::endl, 0; }
    }
    {
        const ull m = *std::min_element(A.begin(), A.end());
        if (m == 1) {
            ull p = 1;
            for (const ull a : A) { p *= (a + 1); }
            p--;
            for (int i = 1; i <= X; i++) { std::cout << (i >= p); }
            return std::cout << std::endl, 0;
        }
    }

    using mint = modint<mod>;
    std::vector<mint> hs(X + 1), ihs(X + 1, 1);
    for (int i = 0; i <= X; i++) { hs[i] = g_rng_xo64(2, mod - 2); }
    for (int i = 0; i <= X; i++) { ihs[i] /= hs[i]; }
    rbstree<ull> res;
    for (const ull a : A) { res.insert_at(0, a); }
    int n     = N;
    mint hash = 1;
    for (const ull a : A) { hash *= hs[a]; }
    ull product = 1;
    for (const ull a : A) { product *= (a + 1); }

    using pll = std::pair<ull, ull>;
    auto pick = [&](int i, int j) -> pll {
        const ull x = res.at(i).data().value;
        const ull y = res.at(j).data().value;
        assert(product % ((x + 1) * (y + 1)) == 0);
        (product /= (x + 1)) /= (y + 1);
        (hash *= ihs[x]) *= ihs[y];
        res.erase_at(j);
        res.erase_at(i);
        n -= 2;
        return {x, y};
    };
    auto insert = [&](const ull z) {
        res.insert_at(n, z);
        hash *= hs[z];
        product *= (z + 1);
        n++;
    };
    auto remove = [&]() {
        const ull z = res.at(n - 1).data().value;
        res.erase_at(n - 1);
        assert(product % (z + 1) == 0);
        (product /= (z + 1));
        hash *= ihs[z];
        n--;
    };
    auto undo = [&](int i, int j, ull x, ull y) {
        (product *= (x + 1)) *= (y + 1);
        (hash *= hs[x]) *= hs[y];
        res.insert_at(i, x);
        res.insert_at(j, y);
        n += 2;
    };
    std::vector<bool> ok(X + 1, false);
    std::set<uint> used;
    auto dfs = [&](auto&& self) -> void {
        if (used.find(hash()) != used.end()) { return; }
        used.insert(hash());
        if (n == 1) {
            ok[res.at(0).data().value] = true;
            return;
        }
        for (int i = 0; i + 1 < n; i++) {
            for (int j = i + 1; j < n; j++) {
                const auto p = pick(i, j);
                const ull x = p.first, y = p.second;
                ull min = (x + 1) * (y + 1) - 1;
                for (ull i = 0;; i++) {
                    const ull z = min + x * i;
                    if (product * (z + 1) > X + 1) { break; }
                    insert(z);
                    self(self);
                    remove();
                }
                for (ull i = 1;; i++) {
                    const ull z = min + y * i;
                    if (product * (z + 1) > X + 1) { break; }
                    insert(z);
                    self(self);
                    remove();
                }
                undo(i, j, x, y);
            }
        }
    };
    dfs(dfs);
    for (int x = 1; x <= X; x++) { std::cout << ok[x]; }
    std::cout << std::endl;
    return 0;
}