#pragma GCC target("avx2")
#pragma GCC optimization("O3")
#pragma GCC optimization("unroll-loops")
#include <algorithm>
#include <assert.h>
#include <bitset>
#include <cfloat>
#include <complex>
#include <deque>
#include <fstream>
#include <functional>
#include <iomanip>
#include <iostream>
#include <limits.h>
#include <list>
#include <map>
#include <math.h>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <string.h>
#include <time.h>
#include <unordered_map>
#include <unordered_set>
#include <vector>
#define rep(i,n) for(int i=0;i<(n);i++)
#define REP(i,n) for(int i=1;i<=(n);i++)
#define int long long
#define ll long long
#define mod (int)1000000007
#define P pair<int,int>
#define prique(T) priority_queue<T,vector<T>,greater<T>>
#define all(V) V.begin(),V.end()
#ifdef int
constexpr int INF = LLONG_MAX / 10;
#else
constexpr int INF = INT_MAX / 10;
#endif
constexpr double eps = 1e-9;
template <class T, class U>
inline bool chmax(T& lhs, const U& rhs) {
	if (lhs < rhs) {
		lhs = rhs;
		return 1;
	}
	return 0;
}
template <class T, class U>
inline bool chmin(T& lhs, const U& rhs) {
	if (lhs > rhs) {
		lhs = rhs;
		return 1;
	}
	return 0;
}
using namespace std;
inline int gcd(int a, int b) {
	while (b) {
		int c = a;
		a = b; b = c % b;
	}
	return a;
}
inline int lcm(int a, int b) {
	return a / gcd(a, b) * b;
}
bool isprime(int n) {
	if (n == 1)return false;
	for (int i = 2; i * i <= n; i++) {
		if (n % i == 0)return false;
	}
	return true;
}
int mypow(int a, int b) {
	if (!b)return 1;
	if (b & 1)return mypow(a, b - 1) * a;
	int memo = mypow(a, b >> 1);
	return memo * memo;
}
int modpow(int a, int b, int m = mod) {
	if (!b)return 1;
	if (b & 1)return modpow(a, b - 1, m) * a % m;
	int memo = modpow(a, b >> 1, m);
	return memo * memo % m;
}
class modInt {
	int value, modulo;
public:
	modInt() : value(0), modulo(mod) { value = 0; }
	template<typename T>
	modInt(T value = 0, int modulo = mod) : value(value), modulo(modulo) {
		if (value < 0) {
			value = -value;
			value %= modulo;
			value = -value;
			value += modulo;
		}
		this->value = value % modulo;
	}
	inline operator int()const { return value; }
	inline modInt& operator+=(modInt x) {
		value += x.value;
		if (value >= modulo)value -= modulo;
		return *this;
	}
	inline modInt& operator++() {
		if (value == modulo - 1)value = 0;
		else value++;
		return *this;
	}
	inline modInt operator-()const {
		return modInt(0) -= *this;
	}
	inline modInt& operator-=(modInt x) {
		value -= x.value;
		if (value < 0)value += modulo;
		return *this;
	}
	inline modInt& operator--() {
		if (value == 0)value = modulo - 1;
		else value--;
		return *this;
	}
	inline modInt& operator*=(modInt x) {
		value = value * x.value % modulo;
		return *this;
	}
	inline modInt& operator/=(modInt x) {
		return operator*=(x.inv());
	}
	inline modInt inv() {
		return modpow(*this, modulo - 2);
	}
	template<typename T> modInt operator+(T x) { return modInt(*this) += x; }
	template<typename T> modInt& operator+=(T x) { return operator+=(modInt(x, modulo)); }
	template<typename T> modInt operator-(T x) { return modInt(*this) -= x; }
	template<typename T> modInt& operator-=(T x) { return operator-=(modInt(x, modulo)); }
	template<typename T> modInt operator*(T x) { return modInt(*this) *= x; }
	template<typename T> modInt& operator*=(T x) { return operator*=(modInt(x, modulo)); }
	template<typename T> modInt operator/(T x) { return modInt(*this) /= x; }
	template<typename T> modInt& operator/=(T x) { return operator/=(modInt(x, modulo)); }
};
istream& operator>>(istream& ist, modInt& x) {
	int a;
	ist >> a;
	x = a;
	return ist;
}
modInt modpow(modInt a, int b) {
	if (!b)return modInt(1);
	if (b & 1)return modpow(a, b - 1) * a;
	modInt memo = modpow(a, b / 2);
	return memo * memo;
}
class Polynomial {
public:
	vector<modInt> vec;
	Polynomial() {}
	Polynomial(vector<modInt> vec) :vec(vec) {}
	modInt get(modInt x) {
		modInt res = 0;
		rep(i, vec.size()) {
			res += modpow(x, i) * vec[i];
		}
		return res;
	}
	Polynomial integrate() {
		vector<modInt> res = vec;
		res.emplace_back();
		for (int i = (int)res.size() - 2; i >= 0; i--)res[i + 1] = res[i] / (i + 1);
		res[0] = 0;
		return Polynomial(res);
	}
};
class func {
public:
	Polynomial first, second;
	modInt third;
	func() {}
	func(Polynomial first, Polynomial second, modInt third) :first(first), second(second), third(third) {}
	void set(Polynomial first, Polynomial second, modInt third) {
		this->first = first;
		this->second = second;
		this->third = third;
	}
	func* operator+=(const func& f) {
		first.vec.resize(max(first.vec.size(), f.first.vec.size()));
		second.vec.resize(max(second.vec.size(), f.second.vec.size()));
		rep(i, f.first.vec.size())first.vec[i] += f.first.vec[i];
		rep(i, f.second.vec.size())second.vec[i] += f.second.vec[i];
		third += f.third;
		return this;
	}
};
func dintegrate(func f, bool flag) {
	Polynomial p = f.first.integrate(), q = f.second.integrate();
	if (!flag) {
		p.vec[0] = f.third;
		return func(p, q, -f.third);
	}
	else {
		p.vec[0] -= p.get(1);
		rep(i, p.vec.size())p.vec[i] = -p.vec[i];
		q.vec[0] -= q.get(1);
		q.vec[0] += f.third;
		rep(i, q.vec.size())q.vec[i] = -q.vec[i];
		return func(p, q, f.third);
	}
}
int n;
string s;
func dp[1010][2][2][2];
modInt fact[1010], inv[1010];
signed main() {
	cin >> n >> s;
	if (s[0] == 'X')dp[0][0][1][0].set(vector<modInt>(), vector<modInt>(), 1);
	else {
		dp[0][0][0][1].set(vector<modInt>(), vector<modInt>(), 1);
		dp[0][1][1][1].set(vector<modInt>(1, 1), vector<modInt>(), modInt(-1));
	}
	REP(i, n - 1) {
		if (s[i] == 'X') {
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][0], 0);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][1], 0);
			dp[i][0][1][0] += dintegrate(dp[i - 1][0][0][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][0][1][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][0][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][1], 1);
		}
		else {
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][0], 0);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][1], 0);
			dp[i][1][1][1] += dintegrate(dp[i - 1][0][1][0], 0);
			dp[i][1][1][1] += dintegrate(dp[i - 1][0][1][1], 0);
			dp[i][0][0][1] += dintegrate(dp[i - 1][0][1][0], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][0][1][1], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][0], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][1], 1);
		}
	}
	fact[0] = 1;
	REP(i, n)fact[i] = fact[i - 1] * i;
	inv[n] = modInt(1) / fact[n];
	for (int i = n - 1; i >= 0; i--)inv[i] = inv[i + 1] * (i + 1);
	modInt a, b, c;
	Polynomial p;

	p = dp[n - 1][0][1][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][0][1][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][0][1][1].third;
	b -= dp[n - 1][0][1][1].third;
	p = dp[n - 1][1][1][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][1][1][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][1][1][1].third;
	b -= dp[n - 1][1][1][1].third;

	rep(i, dp[n - 1][0][1][0].first.vec.size()) {
		a += dp[n - 1][0][1][0].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b -= dp[n - 1][0][1][0].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][0][1][0].second.vec.size()) {
		b += dp[n - 1][0][1][0].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c -= dp[n - 1][0][1][0].second.vec[i] * fact[i] * inv[j];
		}
	}
	a += dp[n - 1][0][1][0].third / 2;
	c -= dp[n - 1][0][1][0].third / 2;

	rep(i, dp[n - 1][1][1][0].first.vec.size()) {
		a += dp[n - 1][1][1][0].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b -= dp[n - 1][1][1][0].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][1][1][0].second.vec.size()) {
		b += dp[n - 1][1][1][0].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c -= dp[n - 1][1][1][0].second.vec[i] * fact[i] * inv[j];
		}
	}
	a += dp[n - 1][1][1][0].third / 2;
	c -= dp[n - 1][1][1][0].third / 2;

	p = dp[n - 1][0][0][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][0][0][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][0][0][1].third;
	b -= dp[n - 1][0][0][1].third;

	p = dp[n - 1][1][0][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][1][0][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][1][0][1].third;
	b -= dp[n - 1][1][0][1].third;

	rep(i, dp[n - 1][0][0][1].first.vec.size()) {
		a -= dp[n - 1][0][0][1].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b += dp[n - 1][0][0][1].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][0][0][1].second.vec.size()) {
		b -= dp[n - 1][0][0][1].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c += dp[n - 1][0][0][1].second.vec[i] * fact[i] * inv[j];
		}
	}
	a -= dp[n - 1][0][0][1].third / 2;
	c += dp[n - 1][0][0][1].third / 2;

	rep(i, dp[n - 1][1][0][1].first.vec.size()) {
		a -= dp[n - 1][1][0][1].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b += dp[n - 1][1][0][1].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][1][0][1].second.vec.size()) {
		b -= dp[n - 1][1][0][1].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c += dp[n - 1][1][0][1].second.vec[i] * fact[i] * inv[j];
		}
	}
	a -= dp[n - 1][1][0][1].third / 2;
	c += dp[n - 1][1][0][1].third / 2;

	cout << a << " " << b << " " << c << endl;
	return 0;
}
### Prompt

Generate a CPP solution to the following problem:

There is a very long bench. The bench is divided into M sections, where M is a very large integer.

Initially, the bench is vacant. Then, M people come to the bench one by one, and perform the following action:

* We call a section comfortable if the section is currently unoccupied and is not adjacent to any occupied sections. If there is no comfortable section, the person leaves the bench. Otherwise, the person chooses one of comfortable sections uniformly at random, and sits there. (The choices are independent from each other).



After all M people perform actions, Snuke chooses an interval of N consecutive sections uniformly at random (from M-N+1 possible intervals), and takes a photo. His photo can be described by a string of length N consisting of `X` and `-`: the i-th character of the string is `X` if the i-th section from the left in the interval is occupied, and `-` otherwise. Note that the photo is directed. For example, `-X--X` and `X--X-` are different photos.

What is the probability that the photo matches a given string s? This probability depends on M. You need to compute the limit of this probability when M goes infinity.

Here, we can prove that the limit can be uniquely written in the following format using three rational numbers p, q, r and e = 2.718 \ldots (the base of natural logarithm):

p + \frac{q}{e} + \frac{r}{e^2}

Your task is to compute these three rational numbers, and print them modulo 10^9 + 7, as described in Notes section.

Constraints

* 1 \leq N \leq 1000
* |s| = N
* s consists of `X` and `-`.

Input

Input is given from Standard Input in the following format:


N
s


Output

Print three rational numbers p, q, r, separated by spaces.

Examples

Input

1
X


Output

500000004 0 500000003


Input

3
---


Output

0 0 0


Input

5
X--X-


Output

0 0 1


Input

5
X-X-X


Output

500000004 0 833333337


Input

20
-X--X--X-X--X--X-X-X


Output

0 0 183703705


Input

100
X-X-X-X-X-X-X-X-X-X--X-X-X-X-X-X-X-X-X-X-X-X-X-X-X--X--X-X-X-X--X--X-X-X--X-X-X--X-X--X--X-X--X-X-X-


Output

0 0 435664291

### Response

```cpp
#pragma GCC target("avx2")
#pragma GCC optimization("O3")
#pragma GCC optimization("unroll-loops")
#include <algorithm>
#include <assert.h>
#include <bitset>
#include <cfloat>
#include <complex>
#include <deque>
#include <fstream>
#include <functional>
#include <iomanip>
#include <iostream>
#include <limits.h>
#include <list>
#include <map>
#include <math.h>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <string.h>
#include <time.h>
#include <unordered_map>
#include <unordered_set>
#include <vector>
#define rep(i,n) for(int i=0;i<(n);i++)
#define REP(i,n) for(int i=1;i<=(n);i++)
#define int long long
#define ll long long
#define mod (int)1000000007
#define P pair<int,int>
#define prique(T) priority_queue<T,vector<T>,greater<T>>
#define all(V) V.begin(),V.end()
#ifdef int
constexpr int INF = LLONG_MAX / 10;
#else
constexpr int INF = INT_MAX / 10;
#endif
constexpr double eps = 1e-9;
template <class T, class U>
inline bool chmax(T& lhs, const U& rhs) {
	if (lhs < rhs) {
		lhs = rhs;
		return 1;
	}
	return 0;
}
template <class T, class U>
inline bool chmin(T& lhs, const U& rhs) {
	if (lhs > rhs) {
		lhs = rhs;
		return 1;
	}
	return 0;
}
using namespace std;
inline int gcd(int a, int b) {
	while (b) {
		int c = a;
		a = b; b = c % b;
	}
	return a;
}
inline int lcm(int a, int b) {
	return a / gcd(a, b) * b;
}
bool isprime(int n) {
	if (n == 1)return false;
	for (int i = 2; i * i <= n; i++) {
		if (n % i == 0)return false;
	}
	return true;
}
int mypow(int a, int b) {
	if (!b)return 1;
	if (b & 1)return mypow(a, b - 1) * a;
	int memo = mypow(a, b >> 1);
	return memo * memo;
}
int modpow(int a, int b, int m = mod) {
	if (!b)return 1;
	if (b & 1)return modpow(a, b - 1, m) * a % m;
	int memo = modpow(a, b >> 1, m);
	return memo * memo % m;
}
class modInt {
	int value, modulo;
public:
	modInt() : value(0), modulo(mod) { value = 0; }
	template<typename T>
	modInt(T value = 0, int modulo = mod) : value(value), modulo(modulo) {
		if (value < 0) {
			value = -value;
			value %= modulo;
			value = -value;
			value += modulo;
		}
		this->value = value % modulo;
	}
	inline operator int()const { return value; }
	inline modInt& operator+=(modInt x) {
		value += x.value;
		if (value >= modulo)value -= modulo;
		return *this;
	}
	inline modInt& operator++() {
		if (value == modulo - 1)value = 0;
		else value++;
		return *this;
	}
	inline modInt operator-()const {
		return modInt(0) -= *this;
	}
	inline modInt& operator-=(modInt x) {
		value -= x.value;
		if (value < 0)value += modulo;
		return *this;
	}
	inline modInt& operator--() {
		if (value == 0)value = modulo - 1;
		else value--;
		return *this;
	}
	inline modInt& operator*=(modInt x) {
		value = value * x.value % modulo;
		return *this;
	}
	inline modInt& operator/=(modInt x) {
		return operator*=(x.inv());
	}
	inline modInt inv() {
		return modpow(*this, modulo - 2);
	}
	template<typename T> modInt operator+(T x) { return modInt(*this) += x; }
	template<typename T> modInt& operator+=(T x) { return operator+=(modInt(x, modulo)); }
	template<typename T> modInt operator-(T x) { return modInt(*this) -= x; }
	template<typename T> modInt& operator-=(T x) { return operator-=(modInt(x, modulo)); }
	template<typename T> modInt operator*(T x) { return modInt(*this) *= x; }
	template<typename T> modInt& operator*=(T x) { return operator*=(modInt(x, modulo)); }
	template<typename T> modInt operator/(T x) { return modInt(*this) /= x; }
	template<typename T> modInt& operator/=(T x) { return operator/=(modInt(x, modulo)); }
};
istream& operator>>(istream& ist, modInt& x) {
	int a;
	ist >> a;
	x = a;
	return ist;
}
modInt modpow(modInt a, int b) {
	if (!b)return modInt(1);
	if (b & 1)return modpow(a, b - 1) * a;
	modInt memo = modpow(a, b / 2);
	return memo * memo;
}
class Polynomial {
public:
	vector<modInt> vec;
	Polynomial() {}
	Polynomial(vector<modInt> vec) :vec(vec) {}
	modInt get(modInt x) {
		modInt res = 0;
		rep(i, vec.size()) {
			res += modpow(x, i) * vec[i];
		}
		return res;
	}
	Polynomial integrate() {
		vector<modInt> res = vec;
		res.emplace_back();
		for (int i = (int)res.size() - 2; i >= 0; i--)res[i + 1] = res[i] / (i + 1);
		res[0] = 0;
		return Polynomial(res);
	}
};
class func {
public:
	Polynomial first, second;
	modInt third;
	func() {}
	func(Polynomial first, Polynomial second, modInt third) :first(first), second(second), third(third) {}
	void set(Polynomial first, Polynomial second, modInt third) {
		this->first = first;
		this->second = second;
		this->third = third;
	}
	func* operator+=(const func& f) {
		first.vec.resize(max(first.vec.size(), f.first.vec.size()));
		second.vec.resize(max(second.vec.size(), f.second.vec.size()));
		rep(i, f.first.vec.size())first.vec[i] += f.first.vec[i];
		rep(i, f.second.vec.size())second.vec[i] += f.second.vec[i];
		third += f.third;
		return this;
	}
};
func dintegrate(func f, bool flag) {
	Polynomial p = f.first.integrate(), q = f.second.integrate();
	if (!flag) {
		p.vec[0] = f.third;
		return func(p, q, -f.third);
	}
	else {
		p.vec[0] -= p.get(1);
		rep(i, p.vec.size())p.vec[i] = -p.vec[i];
		q.vec[0] -= q.get(1);
		q.vec[0] += f.third;
		rep(i, q.vec.size())q.vec[i] = -q.vec[i];
		return func(p, q, f.third);
	}
}
int n;
string s;
func dp[1010][2][2][2];
modInt fact[1010], inv[1010];
signed main() {
	cin >> n >> s;
	if (s[0] == 'X')dp[0][0][1][0].set(vector<modInt>(), vector<modInt>(), 1);
	else {
		dp[0][0][0][1].set(vector<modInt>(), vector<modInt>(), 1);
		dp[0][1][1][1].set(vector<modInt>(1, 1), vector<modInt>(), modInt(-1));
	}
	REP(i, n - 1) {
		if (s[i] == 'X') {
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][0], 0);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][1], 0);
			dp[i][0][1][0] += dintegrate(dp[i - 1][0][0][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][0][1][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][0][1], 1);
			dp[i][0][1][0] += dintegrate(dp[i - 1][1][1][1], 1);
		}
		else {
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][0], 0);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][1], 0);
			dp[i][1][1][1] += dintegrate(dp[i - 1][0][1][0], 0);
			dp[i][1][1][1] += dintegrate(dp[i - 1][0][1][1], 0);
			dp[i][0][0][1] += dintegrate(dp[i - 1][0][1][0], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][0][1][1], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][0], 1);
			dp[i][0][0][1] += dintegrate(dp[i - 1][1][1][1], 1);
		}
	}
	fact[0] = 1;
	REP(i, n)fact[i] = fact[i - 1] * i;
	inv[n] = modInt(1) / fact[n];
	for (int i = n - 1; i >= 0; i--)inv[i] = inv[i + 1] * (i + 1);
	modInt a, b, c;
	Polynomial p;

	p = dp[n - 1][0][1][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][0][1][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][0][1][1].third;
	b -= dp[n - 1][0][1][1].third;
	p = dp[n - 1][1][1][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][1][1][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][1][1][1].third;
	b -= dp[n - 1][1][1][1].third;

	rep(i, dp[n - 1][0][1][0].first.vec.size()) {
		a += dp[n - 1][0][1][0].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b -= dp[n - 1][0][1][0].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][0][1][0].second.vec.size()) {
		b += dp[n - 1][0][1][0].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c -= dp[n - 1][0][1][0].second.vec[i] * fact[i] * inv[j];
		}
	}
	a += dp[n - 1][0][1][0].third / 2;
	c -= dp[n - 1][0][1][0].third / 2;

	rep(i, dp[n - 1][1][1][0].first.vec.size()) {
		a += dp[n - 1][1][1][0].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b -= dp[n - 1][1][1][0].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][1][1][0].second.vec.size()) {
		b += dp[n - 1][1][1][0].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c -= dp[n - 1][1][1][0].second.vec[i] * fact[i] * inv[j];
		}
	}
	a += dp[n - 1][1][1][0].third / 2;
	c -= dp[n - 1][1][1][0].third / 2;

	p = dp[n - 1][0][0][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][0][0][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][0][0][1].third;
	b -= dp[n - 1][0][0][1].third;

	p = dp[n - 1][1][0][1].first.integrate();
	a += p.get(1) - p.get(0);
	p = dp[n - 1][1][0][1].second.integrate();
	b += p.get(1) - p.get(0);
	a += dp[n - 1][1][0][1].third;
	b -= dp[n - 1][1][0][1].third;

	rep(i, dp[n - 1][0][0][1].first.vec.size()) {
		a -= dp[n - 1][0][0][1].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b += dp[n - 1][0][0][1].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][0][0][1].second.vec.size()) {
		b -= dp[n - 1][0][0][1].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c += dp[n - 1][0][0][1].second.vec[i] * fact[i] * inv[j];
		}
	}
	a -= dp[n - 1][0][0][1].third / 2;
	c += dp[n - 1][0][0][1].third / 2;

	rep(i, dp[n - 1][1][0][1].first.vec.size()) {
		a -= dp[n - 1][1][0][1].first.vec[i] * fact[i];
		rep(j, i + 1) {
			b += dp[n - 1][1][0][1].first.vec[i] * fact[i] * inv[j];
		}
	}
	rep(i, dp[n - 1][1][0][1].second.vec.size()) {
		b -= dp[n - 1][1][0][1].second.vec[i] * fact[i];
		rep(j, i + 1) {
			c += dp[n - 1][1][0][1].second.vec[i] * fact[i] * inv[j];
		}
	}
	a -= dp[n - 1][1][0][1].third / 2;
	c += dp[n - 1][1][0][1].third / 2;

	cout << a << " " << b << " " << c << endl;
	return 0;
}
```