#define _USE_MATH_DEFINES
#define _EXT_CODECVT_SPECIALIZATIONS_H 1
#define _EXT_ENC_FILEBUF_H 1
#include <bits/stdc++.h>
#include <bits/extc++.h>
using namespace std;
/*
#include <atcoder/all>
using namespace atcoder;
*/
/*
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
using bll = boost::multiprecision::cpp_int;
using bdouble = boost::multiprecision::number<boost::multiprecision::cpp_dec_float<100>>;
using namespace boost::multiprecision;
*/
//#define int long long
using ll = long long;
//constexpr ll MOD = (ll)1e9 + 7; //primitive root = 5
constexpr ll MOD = 998244353; //primitive root = 3
//INT_MAX = (1<<31)-1 = 2147483647, INT64_MAX = (1LL<<63)-1 = 9223372036854775807
constexpr ll INF = std::numeric_limits<ll>::max() == INT_MAX ? (ll)1e9 + 7 : (ll)1e18;
constexpr double EPS = 1e-9;
constexpr ll dx[4] = {1, 0, -1, 0};
constexpr ll dy[4] = {0, 1, 0, -1};
constexpr ll dx8[8] = {1, 0, -1, 0, 1, 1, -1, -1};
constexpr ll dy8[8] = {0, 1, 0, -1, 1, -1, 1, -1};
#if defined(LOCAL_TEST) || defined(LOCAL_DEV)
	#define BOOST_STACKTRACE_USE_ADDR2LINE
	#define BOOST_STACKTRACE_ADDR2LINE_LOCATION /usr/local/opt/binutils/bin/addr2line
	#define _GNU_SOURCE 1
	#include <boost/stacktrace.hpp>
#endif
#ifdef LOCAL_TEST
	namespace std {
		template <typename T> class dvector : public std::vector<T> {
		public:
			using std::vector<T>::vector;
			template <typename T_ = T, typename std::enable_if_t<std::is_same_v<T_, bool>, std::nullptr_t> = nullptr>
			std::vector<bool>::reference operator[](std::size_t n) {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
			template <typename T_ = T, typename std::enable_if_t<std::is_same_v<T_, bool>, std::nullptr_t> = nullptr>
			const T_ operator[](std::size_t n) const {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
			template <typename T_ = T, typename std::enable_if_t<!std::is_same_v<T_, bool>, std::nullptr_t> = nullptr>
			T_& operator[](std::size_t n) {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
			template <typename T_ = T, typename std::enable_if_t<!std::is_same_v<T_, bool>, std::nullptr_t> = nullptr>
			const T_& operator[](std::size_t n) const {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
		};
		template <typename T, typename Compare = std::less<T>, typename Allocator = std::allocator<T>> class dmultiset : public std::multiset<T,Compare,Allocator> {
		public:
			using std::multiset<T,Compare,Allocator>::multiset;
			const typename std::multiset<T,Compare,Allocator>::iterator erase(const typename std::multiset<T,Compare,Allocator>::iterator it) {
				return std::multiset<T,Compare,Allocator>::erase(it);
			}
			std::size_t erase([[maybe_unused]] const T& x) {
				std::cerr << boost::stacktrace::stacktrace() << '\n'; assert(false);
			}
			std::size_t erase_all_elements(const T& x) {
				return std::multiset<T,Compare,Allocator>::erase(x);
			}
		};
	}
	#define vector dvector
	#define multiset dmultiset
	class SIGFPE_exception : std::exception {};
	class SIGSEGV_exception : std::exception {};
	void catch_SIGFPE([[maybe_unused]] int e) { std::cerr << boost::stacktrace::stacktrace() << '\n'; throw SIGFPE_exception(); }
	void catch_SIGSEGV([[maybe_unused]] int e) { std::cerr << boost::stacktrace::stacktrace() << '\n'; throw SIGSEGV_exception(); }
	signed convertedmain();
	signed main() { signal(SIGFPE, catch_SIGFPE); signal(SIGSEGV, catch_SIGSEGV); return convertedmain(); }
	#define main() convertedmain()
#else
	#define erase_all_elements erase
#endif
#ifdef LOCAL_DEV
	template <typename T1, typename T2> std::ostream& operator<<(std::ostream& s, const std::pair<T1, T2>& p) {
		return s << "(" << p.first << ", " << p.second << ")"; }
	template <typename T, std::size_t N> std::ostream& operator<<(std::ostream& s, const std::array<T, N>& a) {
		s << "{ "; for (std::size_t i = 0; i < N; ++i){ s << a[i] << "\t"; } s << "}"; return s; }
	template <typename T, typename Compare> std::ostream& operator<<(std::ostream& s, const std::set<T, Compare>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template <typename T, typename Compare> std::ostream& operator<<(std::ostream& s, const std::multiset<T, Compare>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template <typename T1, typename T2, typename Compare> std::ostream& operator<<(std::ostream& s, const std::map<T1, T2, Compare>& m) {
		s << "{\n"; for (auto itr = m.begin(); itr != m.end(); ++itr){ s << "\t" << (*itr).first << " : " << (*itr).second << "\n"; } s << "}"; return s; }
	template <typename T> std::ostream& operator<<(std::ostream& s, const std::deque<T>& v) {
		for (std::size_t i = 0; i < v.size(); ++i){ s << v[i]; if (i < v.size() - 1) s << "\t"; } return s; }
	template <typename T> std::ostream& operator<<(std::ostream& s, const std::vector<T>& v) {
		for (std::size_t i = 0; i < v.size(); ++i){ s << v[i]; if (i < v.size() - 1) s << "\t"; } return s; }
	template <typename T> std::ostream& operator<<(std::ostream& s, const std::vector<std::vector<T>>& vv) {
		s << "\\\n"; for (std::size_t i = 0; i < vv.size(); ++i){ s << vv[i] << "\n"; } return s; }
	template <typename T, std::size_t N, typename std::enable_if_t<!std::is_same_v<T, char>, std::nullptr_t> = nullptr> std::ostream& operator<<(std::ostream& s, const T (&v)[N]) {
		for (std::size_t i = 0; i < N; ++i){ s << v[i]; if (i < N - 1) s << "\t"; } return s; }
	template <typename T, std::size_t N, std::size_t M, typename std::enable_if_t<!std::is_same_v<T, char>, std::nullptr_t> = nullptr> std::ostream& operator<<(std::ostream& s, const T (&vv)[N][M]) {
		s << "\\\n"; for (std::size_t i = 0; i < N; ++i){ s << vv[i] << "\n"; } return s; }
	#if __has_include(<ext/pb_ds/assoc_container.hpp>)
		template <typename Key, typename Compare> std::ostream& operator<<(std::ostream& s, const __gnu_pbds::tree<Key, __gnu_pbds::null_type, Compare, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>& se) {
			s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
		template <typename Key, typename T, typename Hash> std::ostream& operator<<(std::ostream& s, const __gnu_pbds::gp_hash_table<Key, T, Hash>& m) {
			s << "{\n"; for (auto itr = m.begin(); itr != m.end(); ++itr){ s << "\t" << (*itr).first << " : " << (*itr).second << "\n"; } s << "}"; return s; }
	#endif
	void debug_impl() { std::cerr << '\n'; }
	template <typename Head, typename... Tail> void debug_impl(Head& head, Tail&... tail) { std::cerr << " " << head << (sizeof...(tail) ? "," : ""); debug_impl(tail...); }
	template <typename Head, typename... Tail> void debug_impl(const Head& head, const Tail&... tail) { std::cerr << " " << head << (sizeof...(tail) ? "," : ""); debug_impl(tail...); }
	#define debug(...) do { std::cerr << ":" << __LINE__ << " (" << #__VA_ARGS__ << ") ="; debug_impl(__VA_ARGS__); } while (false)
	constexpr inline long long prodlocal([[maybe_unused]] long long prod, [[maybe_unused]] long long local) { return local; }
#else
	#define debug(...) do {} while (false)
	constexpr inline long long prodlocal([[maybe_unused]] long long prod, [[maybe_unused]] long long local) { return prod; }
#endif
#define repoverload3(_1, _2, _3, name, ...) name
#define rep3(i, a, b) for(ll i=(a), i##_length=(b); i<i##_length; ++i)
#define rep2(i, n) rep3(i, 0, n)
#define rep1(n) rep3(i, 0, n)
#define rep(...) repoverload3(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repeq3(i, a, b) rep3(i, (a)+1, (b)+1)
#define repeq2(i, n) rep3(i, 1, (n)+1)
#define repeq1(n) rep3(i, 1, (n)+1)
#define repeq(...) repoverload3(__VA_ARGS__, repeq3, repeq2, repeq1)(__VA_ARGS__)
#define rrep3(i, a, b) for(ll i=(b)-1; i>=(a); --i)
#define rrep2(i, n) rrep3(i, 0, n)
#define rrep1(n) rrep3(i, 0, n)
#define rrep(...) repoverload3(__VA_ARGS__, rrep3, rrep2, rrep1)(__VA_ARGS__)
#define rrepeq3(i, a, b) rrep3(i, (a)+1, (b)+1)
#define rrepeq2(i, n) rrep3(i, 1, (n)+1)
#define rrepeq1(n) rrep3(i, 1, (n)+1)
#define rrepeq(...) repoverload3(__VA_ARGS__, rrepeq3, rrepeq2, rrepeq1)(__VA_ARGS__)
#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)
void p() { std::cout << '\n'; }
template <typename Head, typename... Tail> void p(Head& head, Tail&... tail) { std::cout << head << (sizeof...(tail) ? " " : ""); p(tail...); }
template <typename Head, typename... Tail> void p(const Head& head, const Tail&... tail) { std::cout << head << (sizeof...(tail) ? " " : ""); p(tail...); }
template <typename T> inline void pv(const T& v) { for(ll i=0, N=std::size(v); i<N; i++) std::cout << v[i] << " \n"[i==N-1]; }
template <typename T> inline bool chmax(T& a, T b) { return a < b && (a = b, true); }
template <typename T> inline bool chmin(T& a, T b) { return a > b && (a = b, true); }
template <typename T> inline void uniq(T& v) { std::sort(v.begin(), v.end()); v.erase(std::unique(v.begin(), v.end()), v.end()); }
template <typename T> inline ll sz(const T& v) { return std::size(v); }
template <typename T, std::size_t N> std::vector<T> make_vector_impl(std::vector<ll>& sizes, typename std::enable_if<(N==1), const T&>::type x) { return std::vector<T>(sizes.front(),x); }
template <typename T, std::size_t N> auto make_vector_impl(std::vector<ll>& sizes, typename std::enable_if<(N>1), const T&>::type x) { ll size=sizes.back(); sizes.pop_back(); return std::vector<decltype(make_vector_impl<T,N-1>(sizes,x))>(size,make_vector_impl<T,N-1>(sizes,x)); }
template <typename T, std::size_t N> auto make_vector(const ll (&sizes)[N], const T& x=T()) { std::vector<ll> s(N); for(std::size_t i=0; i<N; ++i)s[i]=sizes[N-1-i]; return make_vector_impl<T,N>(s,x); }
#if __has_include(<ext/pb_ds/assoc_container.hpp>)
	class custom_hash {
	public:
		static uint64_t splitmix64(uint64_t x) {
			x += 0x9e3779b97f4a7c15, x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9, x = (x ^ (x >> 27)) * 0x94d049bb133111eb; return x ^ (x >> 31);
		}
		size_t operator()(uint64_t x) const {
			static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count(); return splitmix64(x + FIXED_RANDOM);
		}
	};
	template <typename Key, typename Mapped, typename Hash = custom_hash, typename std::enable_if_t<std::is_integral_v<Key>, std::nullptr_t> = nullptr> class fmap : public __gnu_pbds::gp_hash_table<Key, Mapped, Hash> {
	public:
		using __gnu_pbds::gp_hash_table<Key, Mapped, Hash>::gp_hash_table;
		template <typename T> fmap(std::initializer_list<std::initializer_list<T>> il) : __gnu_pbds::gp_hash_table<Key, Mapped, Hash>() {
			for (auto&& x : il) __gnu_pbds::gp_hash_table<Key, Mapped, Hash>::insert(std::pair<Key, Mapped>(*x.begin(), *(x.begin() + 1)));
		}
		template <typename T> ll count(const T& x) const {
			return __gnu_pbds::gp_hash_table<Key, Mapped, Hash>::find(x) != __gnu_pbds::gp_hash_table<Key, Mapped, Hash>::end();
		}
	};
#else
	template <typename Key, typename Mapped> using fmap = std::map<Key, Mapped>;
#endif
template <typename... T> class zip_iterator_base : std::iterator<std::forward_iterator_tag, std::tuple<decltype(*std::declval<T>().begin())...>> {
public:
	ll idx_;
	std::tuple<decltype(std::declval<T>().begin())...> iters_;
	template <std::size_t... I> auto deref(std::index_sequence<I...>) const { return typename zip_iterator_base::value_type{*std::get<I>(iters_)...}; }
	template <std::size_t... I> void increment(std::index_sequence<I...>) { [[maybe_unused]] auto l = {(++std::get<I>(iters_), 0)...}; }
	explicit zip_iterator_base(decltype(iters_) iters) : idx_(0), iters_{std::move(iters)} {}
	zip_iterator_base& operator++() { ++idx_; increment(std::index_sequence_for<T...>{}); return *this; }
	zip_iterator_base operator++(int) { auto saved{*this}; ++idx_; increment(std::index_sequence_for<T...>{}); return saved; }
	bool operator!=(const zip_iterator_base& other) const { return iters_ != other.iters_; }
};
template <typename... T> class zip_iterator : public zip_iterator_base<T...> {
public:
	using zip_iterator_base<T...>::zip_iterator_base;
	auto operator*() const { return this->deref(std::index_sequence_for<T...>{}); }
};
template <typename... T> class zipindex_iterator : public zip_iterator_base<T...> {
public:
	using zip_iterator_base<T...>::zip_iterator_base;
	auto operator*() const { return std::tuple_cat(std::make_tuple(this->idx_), this->deref(std::index_sequence_for<T...>{})); }
};
template <typename IteratorType, typename... T> class zip_helper {
public:
	zip_helper(T&... seqs) : begin_{std::make_tuple(seqs.begin()...)}, end_{std::make_tuple(seqs.end()...)} {}
	IteratorType begin() const { return begin_; }
	IteratorType end() const { return end_; }
	IteratorType begin_, end_;
};
template <typename... T> auto zip(T&&... seqs) { return zip_helper<zip_iterator<T...>, T...>{seqs...}; }
template <typename... T> auto zipindex(T&&... seqs) { return zip_helper<zipindex_iterator<T...>, T...>{seqs...}; }

/*-----8<-----template-----8<-----*/

//[lib]unionfind.cpp
class UnionFind {
public:
	vector<ll> v;
	UnionFind() = default;
	UnionFind(size_t size) : v(size, -1) {}

	ll root(ll x) {
		return (v[x] < 0 ? x : v[x] = root(v[x]));
	}

	bool is_root(ll x) {
		return x == root(x);
	}

	bool is_same(ll x, ll y) {
		return root(x) == root(y);
	}

	bool unite(ll x, ll y) {
		x = root(x);
		y = root(y);
		if (x == y) return false;
		if (v[x] > v[y]) swap(x, y);
		v[x] += v[y];
		v[y] = x;
		return true;
	}

	ll size(ll x) {
		return -v[root(x)];
	}
};
std::ostream& operator<<(std::ostream& s, const UnionFind& uf) { 
	for (size_t i = 0; i < uf.v.size(); ++i){ s << uf.v[i]; if (i < uf.v.size() - 1) s << "\t"; } return s;
}

//[lib](internal)edge.cpp
using EdgeCostType = ll;
using usize = ll;
template <class T> class edge_type {
public:
	usize from, to;
	T cost;
	edge_type() {}
	edge_type(usize from, usize to, T cost) : from(from), to(to), cost(cost) {}
	bool operator<(const edge_type& r) const { return r.cost < cost; }
};
using Edge = edge_type<EdgeCostType>;
ostream& operator<<(ostream& s, const Edge& e) {
	s << "{ " << e.from << " -> " << e.to << ", " << e.cost << " }";
	return s;
}
inline void addedge(vector<vector<Edge>>& g, usize from, usize to, EdgeCostType cost) {
	g[from].emplace_back(from, to, cost);
	g[to].emplace_back(to, from, cost);
}
//最短路木の親頂点を元にstart->goalの経路を作成
vector<ll> buildPath(const vector<ll> &prev, ll goal) {
	vector<ll> path;
	for (ll u = goal; u >= 0; u = prev[u])
		path.push_back(u);
	reverse(path.begin(), path.end());
	return path;
}

//[lib]dijkstra.cpp
//[depends on](internal)edge.cpp
template <class T> class fibonacci_heap {
	class node_type;
	using node_ptr = node_type *;
	class node_type {
	public:
		node_ptr parent;
		node_ptr child;
		node_ptr left;
		node_ptr right;
		usize rank;
		bool mark;
		T key;
		usize prev;

		node_type()
				: parent(nullptr), child(nullptr), left(nullptr), right(nullptr),
					rank(0), mark(false), key(std::numeric_limits<T>::max()), prev(-1) {}
	};

	vector<node_type> nodes;
	node_ptr root;
	vector<node_ptr> table;

public:
	fibonacci_heap(const usize n)
			: nodes(n), root(nullptr),
				table(std::ceil(std::log(n + 1) * 2.08), nullptr) {}

	bool empty() const { return root == nullptr; }

	edge_type<T> pop() {
		edge_type<T> ret = {root->prev, static_cast<usize>(root - nodes.data()), root->key};

		usize max = 0;
		const auto push = [&](node_ptr v) -> void {
			while (true) {
				node_ptr u = table[v->rank];
				if (u == nullptr) {
					table[v->rank] = v;
					break;
				}
				table[v->rank] = nullptr;
				if (u->key < v->key) {
					std::swap(u, v);
				}
				const node_ptr c = v->child;
				if (c == nullptr) {
					u->left = u;
					u->right = u;
					v->child = u;
				} else {
					u->left = c->left;
					u->right = c;
					c->left->right = u;
					c->left = u;
				}
				u->parent = v;
				v->rank += 1;
			}
			max = std::max(max, v->rank + 1);
		};

		{
			node_ptr v = root->right;
			while (v != root) {
				const node_ptr next = v->right;
				push(v);
				v = next;
			}
		}
		if (root->child != nullptr) {
			node_ptr v = root->child;
			do {
				const node_ptr next = v->right;
				v->mark = false;
				push(v);
				v = next;
			} while (v != root->child);
		}
		root = nullptr;

		for (usize i = 0; i != max; i += 1) {
			const node_ptr v = table[i];
			if (v == nullptr) {
				continue;
			}
			table[i] = nullptr;
			v->parent = nullptr;
			if (root == nullptr) {
				root = v;
				v->left = v;
				v->right = v;
			} else {
				v->left = root->left;
				v->right = root;
				root->left->right = v;
				root->left = v;
				if (root->key > v->key) {
					root = v;
				}
			}
		}

		return ret;
	}

	void update_key(const usize v_, const T key, const usize prev) {
		node_ptr v = &nodes[v_];

		if (v->key <= key) {
			return;
		}
		v->key = key;
		v->prev = prev;

		if (v->left == nullptr) {
			if (root == nullptr) {
				v->left = v;
				v->right = v;
				root = v;
			} else {
				v->left = root->left;
				v->right = root;
				root->left->right = v;
				root->left = v;
				if (key < root->key) {
					root = v;
				}
			}
			return;
		}

		if (v->parent == nullptr) {
			if (key < root->key) {
				root = v;
			}
			return;
		} else {
			if (v->parent->key <= key) {
				return;
			}
		}

		while (true) {
			const node_ptr p = v->parent;
			v->left->right = v->right;
			v->right->left = v->left;
			v->parent = nullptr;
			p->rank -= 1;
			if (p->child == v) {
				if (p->rank == 0) {
					p->child = nullptr;
				} else {
					p->child = v->right;
				}
			}

			v->left = root->left;
			v->right = root;
			root->left->right = v;
			root->left = v;
			v->mark = false;

			v = p;
			if (v->parent == nullptr) {
				break;
			}
			if (!v->mark) {
				v->mark = true;
				break;
			}
		}
		if (root->key > key) {
			root = &nodes[v_];
		}
	}
};
/*
計算量：O(E+VlogV)
引数
　g:探索するグラフ
　start:探索するスタートノード番号
戻り値
　dist:スタートノードから各頂点までの距離
　prev:最短路木の親頂点
*/
void dijkstra(const vector<vector<Edge>> &g, ll start, vector<EdgeCostType> &dist, vector<ll> &prev) {
	dist.assign(g.size(), INF); dist[start] = 0;
	prev.assign(g.size(), -1);
	fibonacci_heap<EdgeCostType> heap(g.size());
	heap.update_key(start, 0, -1);
	while (!heap.empty()) {
		const auto top = heap.pop();
		dist[top.to] = top.cost;
		prev[top.to] = top.from;
		for (const auto &edge : g[top.to]) {
			heap.update_key(edge.to, top.cost + edge.cost, edge.from);
		}
	}
}
template <class Func>
void dijkstra(const ll gsize, const Func &gfunc, ll start, vector<EdgeCostType> &dist, vector<ll> &prev) {
	dist.assign(gsize, INF); dist[start] = 0;
	prev.assign(gsize, -1);
	fibonacci_heap<EdgeCostType> heap(gsize);
	heap.update_key(start, 0, -1);
	while (!heap.empty()) {
		const auto top = heap.pop();
		dist[top.to] = top.cost;
		prev[top.to] = top.from;
		gfunc(top.to, [&](const usize nextv, const EdgeCostType nextcost) {
			heap.update_key(nextv, top.cost + nextcost, top.to);
		});
	}
}

//[lib]dijkstra区間.cpp
//[depends on]dijkstra.cpp
//セグ木の形にして区間に辺を張る
class Graph {
public:
	ll N,sz;
	//この g にグラフを作るので、これを使ってdijkstraしてください
	vector<vector<Edge>> g;

	Graph (ll N) : N(N) {
		sz = 1;
		while(sz < N) sz <<= 1;
		g.resize(sz*3-2);
		ll diff=sz;
		for(ll i=0; i<sz/2-1; i++){
			g[i*2+1+diff].emplace_back(i*2+1+diff, i+diff, 0);
			g[i*2+2+diff].emplace_back(i*2+2+diff, i+diff, 0);
		}
		for(ll i=sz*3/2-1; i<sz*2-1; i++){
			ll t=(i-(sz*3/2-1))*2;
			g[t].emplace_back(t, i, 0);
			g[t+1].emplace_back(t+1, i, 0);
		}
		diff=sz*2-1;
		for(ll i=0; i<sz/2-1; i++){
			g[i+diff].emplace_back(i+diff, i*2+1+diff, 0);
			g[i+diff].emplace_back(i+diff, i*2+2+diff, 0);
		}
		for(ll i=sz*5/2-2; i<sz*3-2; i++){
			ll t=(i-(sz*5/2-2))*2;
			g[i].emplace_back(i, t, 0);
			g[i].emplace_back(i, t+1, 0);
		}
	};

	//[froml, fromr) -> [tol, tor) に 距離costの辺を張る
	void add_edge(ll froml, ll fromr, ll tol, ll tor, EdgeCostType cost){
		if(froml>=fromr || tol>=tor) return;
		ll from = g.size();
		ll to = from + 1;
		g.emplace_back();
		g.emplace_back();
		g[from].emplace_back(from, to, cost);
		recursive(froml, fromr, 0, 0, sz, -1, from, sz);
		recursive(tol, tor, 0, 0, sz, to, -1, sz*2-1);
		debug(sz);
		//connect(froml, fromr, -1, from, sz);
		//connect(tol, tor, to, -1, sz*2-1);
	}
	void connect(ll l, ll r, ll from, ll to, ll diff) {
		bool isfirst = true;
		for (l += sz, r += sz; l < r; l >>= 1, r >>= 1, isfirst = false) {
			if(l&1) {
				
				//res.push_back(l++);
				ll v = isfirst ? l-sz : l+diff-1;
				debug(l, v);
				connect_edge(from, to, v);
				l++;
			}
			if(r&1) {
				--r;
				ll v = isfirst ? r-sz : r+diff-1;
				connect_edge(from, to, v);
				debug(r, v);
				// res.push_back(--r);
			}
		}
	}
	void connect_edge(ll from, ll to, ll v){
		debug(v, from, to);
		if (from != -1) {
			g[from].emplace_back(from, v, 0);
		} else {
			g[v].emplace_back(v, to, 0);
		}
	}
	// 要求区間 [a, b) 中の要素の最小値を答える
	// k := 自分がいるノードのインデックス
	// 対象区間は [l, r) にあたる
	void recursive(ll a, ll b, ll k, ll l, ll r, ll from, ll to, ll diff) {
		// 要求区間と対象区間が交わらない -> 適当に返す
		if(r <= a || b <= l) return;

		// 要求区間が対象区間を完全に被覆 -> 対象区間を答えの計算に使う
		if(a <= l && r <= b) {
			ll v = r - l > 1 ? k + diff : k - sz + 1;
			debug(k,v,from,to);
			if (from != -1) {
				g[from].emplace_back(from, v, 0);
			} else {
				g[v].emplace_back(v, to, 0);
			}
			return;
		}

		// 要求区間が対象区間の一部を被覆 -> 子について探索を行う
		// 左側の子を vl ・ 右側の子を vr としている
		// 新しい対象区間は、現在の対象区間を半分に割ったもの
		recursive(a, b, 2*k+1, l, (l+r)/2, from, to, diff);
		recursive(a, b, 2*k+2, (l+r)/2, r, from, to, diff);
	}
};

/*-----8<-----library-----8<-----*/

void solve() {
	ll N, A, B;
	cin >> N >> A >> B;
	vector<ll> a(N);
	rep(i, N) cin >> a[i];

	Graph graph(N);
	rep(i,N){
		{
			ll l = lower_bound(all(a), a[i] - B) - a.begin();
			ll r = upper_bound(all(a), a[i] - A) - a.begin();
			graph.add_edge(i, i + 1, l, r, 1);
			debug("----");
		}
		{
			ll l = lower_bound(all(a), a[i] + A) - a.begin();
			ll r = upper_bound(all(a), a[i] + B) - a.begin();
			graph.add_edge(i, i + 1, l, r, 1);
			debug("----!!!!");
		}
	}

	auto& g = graph.g;
	UnionFind uf(g.size());
	vector<ll> seen(g.size(), INF);
	rep(k,N){
		if (seen[k] != INF) continue;
		vector<ll> v;
		queue<ll> que;
		ll startpos = k;
		que.push(startpos);
		seen[k] = 0;
		while(!que.empty()) {
			auto x = que.front();
			if (x < N) v.push_back(x);
			que.pop();
			for(auto&& e:g[x]){
				// seen[nx][ny]の書き換えはここでやること
				if(chmin(seen[e.to], seen[x] + 1)) {
					que.push(e.to);
				}
			}
		}
		rep(i,v.size()-1){
			uf.unite(v[i], v[i + 1]);
		}
	}

	rep(i,N){
		p(uf.size(i));
	}

	std::cout << std::flush;
	_Exit(0);
}

signed main() {
#ifndef LOCAL_DEV
	std::cin.tie(nullptr);
	std::ios::sync_with_stdio(false);
#endif
	//ll Q; cin >> Q; while(Q--)solve();
	solve();
	return 0;
}