#include <bits/stdc++.h>
#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)
template<typename T>
T make_v(const T v) { return v; }
template<typename... Args>
auto make_v(const std::size_t size, Args... args) { return std::vector<decltype(make_v(args...))>(size, make_v(args...)); }


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_splaytree
{
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; }
        bool has_parent() const { return p; }
        const node& left() const { return deptr(l); }
        const node& right() const { return deptr(r); }
        const node& parent() const { return deptr(p); }
        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; }
            ptr lp2    = kth_ptr(tp2, 0);
            return tp2 = (lp2->splay(), lp2), tp2->push_down(), tp2->l = tp1, tp1->p = tp2, tp2->pull_up(), tp2;
        }
        static std::pair<ptr, ptr> split_at(tree tp, const usize pos) { return split(tp, kth_ptr(tp, pos)); }
        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) { return split(tp, lower_bound(tp, key)); }
        template<typename Key>
        static std::pair<ptr, ptr> split_upper(tree tp, const Key& key) { return split(tp, upper_bound(tp, key)); }
        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; }
        bool is_root() const { return (not p) or (p->l != this and p->r != this); }
        int is_left() const { return is_root() ? 0 : (p->l == this ? 1 : -1); }
        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)); }
        void rot()
        {
            ptr pp        = p->p;
            const int isl = p->is_left();
            if (is_left() == 1) {
                p->l = r, (r ? r->p : p) = p, r = p, p->p = this;
            } else {
                p->r = l, (l ? l->p : p) = p, l = p, p->p = this;
            }
            p->pull_up(), pull_up();
            p = pp;
            if (isl == 0) { return; }
            (isl == 1 ? pp->l : pp->r) = this, pp->pull_up();
        }
        void splay()
        {
            while (not is_root()) { (p->is_root() ? rot() : is_left() == p->is_left() ? (p->rot(), rot()) : (rot(), rot())); }
        }
        static std::pair<ptr, ptr> split(tree tp, ptr p)
        {
            if (not p) { return {tp, nullptr}; }
            tp     = (p->splay(), p);
            ptr lp = tp->l;
            if (not lp) { return std::make_pair(nullptr, tp); }
            return tp->push_down(), lp->p = nullptr, tp->l = nullptr, tp->pull_up(), std::make_pair(lp, tp);
        }
        static ptr kth_ptr(const tree tp, usize k)
        {
            for (ptr p = tp; p;) {
                p->push_down();
                const usize ind = size_of(p->l);
                if (ind == k) { return p; }
                p = (k < ind ? p->l : (k -= (ind + 1), p->r));
            }
            return nullptr;
        }
        template<typename Key>
        static ptr lower_bound(const tree tp, const Key& key)
        {
            static typename NodeData::comp_type comp{};
            ptr ans = nullptr;
            for (ptr p = tp; p;) { p->push_down(), p = (comp(p->data().key, key) ? p->r : (ans = p, p->l)); }
            return ans;
        }
        template<typename Key>
        static ptr upper_bound(const tree tp, const Key& key)
        {
            static typename NodeData::comp_type comp{};
            ptr ans = nullptr;
            for (ptr p = tp; p;) { p->push_down(), p = (comp(key, p->data().key) ? (ans = p, p->l) : p->r); }
            return ans;
        }
        static node empty_node;  // inline変数を使いたいでござる
        NodeData node_data;
        ptr l = nullptr, r = nullptr, p = nullptr;
    };
    using ptr       = node*;
    using const_ptr = const node* const;
    ptr root        = nullptr;

public:
    base_splaytree() {}
    base_splaytree(const ptr r) : root{r} {}
    template<typename... Args>
    base_splaytree(Args... args) : root{new node{args...}} {}
    bool empty() const { return not root; }
    usize size() const { return root ? root->sz : 0UL; }
    const node& top() const { return node::deptr(root); }
    const node at(const usize pos) { return fold_range(pos, pos + 1); }
    const 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>
    const 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>
    const 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>
    const node fold_upper(const Key& key)
    {
        auto ts        = node::split_upper(root, key);
        const node ans = node::deptr(ts.second);
        return root    = node::merge(ts.first, ts.second), ans;
    }
    template<typename Value>
    base_splaytree& 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_splaytree& 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_splaytree& 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_splaytree& 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_splaytree& merge(base_splaytree&& t) { return root = node::merge(root, t.root), *this; }
    base_splaytree split_at(const usize pos)
    {
        auto ts     = node::split_at(root, pos);
        return root = ts.first, base_splaytree(ts.second);
    }
    std::pair<base_splaytree, base_splaytree> 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_splaytree(std::get<1>(ts)), base_splaytree(std::get<2>(ts)));
    }
    base_splaytree& 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_splaytree& insert_at(const usize pos, base_splaytree&& 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_splaytree split_lower(const Key& key)
    {
        auto ts     = node::split_lower(root, key);
        return root = ts.first, base_splaytree(ts.second);
    }
    template<typename Key>
    base_splaytree split_upper(const Key& key)
    {
        auto ts     = node::split_upper(root, key);
        return root = ts.first, base_splaytree(ts.second);
    }
    template<typename Key>
    std::pair<base_splaytree, base_splaytree> 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_splaytree(std::get<1>(ts)), base_splaytree(std::get<2>(ts)));
    }
    base_splaytree& 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() const
    {
        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_splaytree<NodeData>::node base_splaytree<NodeData>::node::empty_node = node{};
template<typename Value>
using splaytree = base_splaytree<bbst_node::value_node<Value>>;
template<typename Key, typename Comp = std::less<Key>>
using mset_splaytree = base_splaytree<bbst_node::key_node<Key, bbst_node::node, Comp>>;
template<typename Key, typename Value, typename Comp = std::less<Key>>
using mmap_splaytree = base_splaytree<bbst_node::key_node<Key, bbst_node::value_node<Value>, Comp>>;
template<typename ValueMonoid>
using merge_splaytree = base_splaytree<bbst_node::merge_node<ValueMonoid>>;
template<typename MonoidAct>
using lazy_splaytree = base_splaytree<bbst_node::lazy_node<MonoidAct>>;



/**
 * 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;
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>>;

class segments
{
private:
    using P = std::pair<usize, usize>;
    const usize ceil;
    std::vector<P> rs;
    const usize num;  // 区間0は存在しないので、実際は区間1,2,...,num-1

public:
    segments(const usize sz) : ceil{ceil2(sz)}, rs(ceil << 1, P{0, 0}), num{ceil << 1}
    {
        for (usize sz = 1; sz <= ceil; sz <<= 1) {
            const usize len = ceil / sz;
            for (usize j = sz; j < (sz << 1); j++) { rs[j] = {len * (j - sz), len * (j - sz + 1)}; }
        }
    }
    std::vector<usize> under(usize l, usize r) const
    {
        assert(l < r), assert(r <= ceil);
        std::vector<usize> lind, rind;
        for (l += ceil, r += ceil; l < r; l >>= 1, r >>= 1) {
            if (l & 1) { lind.push_back(l++); }
            if (r & 1) { rind.push_back(--r); }
        }
        for (; not rind.empty(); rind.pop_back()) { lind.push_back(rind.back()); }
        return lind;
    }
    std::vector<usize> over(const usize a) const
    {
        assert(a < ceil);
        std::vector<usize> ans;
        for (usize i = a + ceil; i >= 1; i >>= 1) { ans.push_back(i); }
        std::reverse(ans.begin(), ans.end());
        return ans;
    }
    usize max_index() const { return num; }
    const P& operator[](const usize i) const
    {
        assert(i >= 1), assert(i < 2 * ceil);
        return rs[i];
    }
};
template<typename Element>
struct plus
{
    using element_type  = Element;
    using operator_type = element_type;
    plus()              = delete;
    static operator_type compose(const operator_type& a, const operator_type& b) { return a + b; }
    static constexpr operator_type id() { return operator_type{}; }
};
template<typename Element>
struct sum
{
    using element_type = Element;
    using value_type   = element_type;
    sum()              = delete;
    static value_type merge(const value_type& a, const value_type& b) { return a + b; }
    static constexpr value_type id() { return value_type{}; }
};
template<typename ValueElement, typename OperatorElement>
struct sum_plus
{
    using value_element_type    = ValueElement;
    using operator_element_type = OperatorElement;
    using value_monoid_type     = sum<value_element_type>;
    using operator_monoid_type  = plus<operator_element_type>;
    using value_type            = typename value_monoid_type::value_type;
    using operator_type         = typename operator_monoid_type::operator_type;
    sum_plus()                  = delete;
    template<typename Ind>
    static value_type apply(const operator_type& f, const value_type& x, const Ind l) { return x + static_cast<value_type>(l) * static_cast<value_type>(f); }
};
int main()
{
    constexpr ull MAX = 1ULL << 40;
    using node        = bbst_node::key_node<ull, bbst_node::lazy_node<sum_plus<ull, ull>>, std::less<ull>>;
    using tree        = base_splaytree<node>;
    //using tree        = base_rbstree<node>;
    const auto [n, q] = reads<usize, usize>();
    const auto m      = ceil2(n);
    auto a            = read<ull>(n);
    std::vector<tree> sorted(2 * m);
    segments segs{n};
    for (usize i = 1; i < 2 * m; i++) {
        const auto [l, r] = segs[i];
        for (usize j = l; j < std::min(r, n); j++) { sorted[i].insert({a[j], a[j]}); }
    }
    ull s               = 0;
    constexpr ull pmask = (1ULL << 16) - 1;
    constexpr ull vmask = (1ULL << 40) - 1;
    for (usize i = 0; i < q; i++) {
        const usize t = read<usize>();
        if (t == 1) {
            const usize sp  = s & pmask;
            const ull sv    = s & vmask;
            const auto x    = (read<usize>() ^ (sp)) - 1;
            const auto y    = (read<ull>() ^ sv);
            const auto inds = segs.over(x);
            for (const usize i : inds) {
                auto& lst       = sorted[i];
                auto [mst, rst] = lst.split_key_range(a[x], a[x]);
                auto mst2       = mst.split_at(1);
                mst2.merge(std::move(rst));
                lst.merge(std::move(mst2));
                lst.insert({y, y});
            }
            a[x] = y;
        } else {
            const usize sp = s & pmask;
            auto l = (read<usize>() ^ (sp)), r = (read<usize>() ^ (sp));
            if (l > r) { std::swap(l, r); }
            l--;
            const usize sz  = r - l;
            const auto inds = segs.under(l, r);
            auto less       = [&](const ll x) -> usize {
                usize ans = 0;
                for (const usize i : inds) { ans += sorted[i].fold_lower(x).data().sz; }
                return ans;
            };
            ll inf = -1LL, sup = MAX + 1LL;
            while (sup - inf > 1) {
                const ll mid                           = (inf + sup) / 2LL;
                (less(mid) < (sz + 1) / 2 ? inf : sup) = mid;
            }
            const ll m = inf;
            ll ans     = 0;
            for (const usize i : inds) {
                auto& lst     = sorted[i];
                auto rst      = sorted[i].split_lower(m);
                const usize l = lst.top().data().sz;
                const usize r = rst.top().data().sz;
                const ll lsum = (ll)l * m - lst.top().data().merged;
                const ll rsum = rst.top().data().merged - (ll)r * m;
                ans += (lsum + rsum);
                lst.merge(std::move(rst));
            }
            s ^= ans;
            std::cout << ans << std::endl;
        }
    }
    return 0;
}