Problem:
Score : 800 points

Problem Statement

You are given integers N,\ A and B.
Determine if there exists a permutation (P_0,\ P_1,\ ...\ P_{2^N-1}) of (0,\ 1,\ ...\ 2^N-1) that satisfies all of the following conditions, and create one such permutation if it exists.

P_0=A

P_{2^N-1}=B

For all 0 \leq i < 2^N-1, the binary representations of P_i and P_{i+1} differ by exactly one bit.

Constraints

1 \leq N \leq 17

0 \leq A \leq 2^N-1

0 \leq B \leq 2^N-1

A \neq B

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B

Output

If there is no permutation that satisfies the conditions, print NO.

If there is such a permutation, print YES in the first line.
Then, print (P_0,\ P_1,\ ...\ P_{2^N-1}) in the second line, with spaces in between.
If there are multiple solutions, any of them is accepted.

Sample Input 1

2 1 3

Sample Output 1

YES
1 0 2 3

The binary representation of P=(1,0,2,3) is (01,00,10,11), where any two adjacent elements differ by exactly one bit.

Sample Input 2

3 2 1

Sample Output 2

NO

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#include <array>
using namespace std;
using ULL = unsigned long long;
using UL = unsigned;
using LL = long long;
#define rep(i, n) for(UL i = 0; i < (n); i++)

struct Problem {

	vector<UL> Solve2(UL N, UL B) {
		if (N == 1) {
			if ((B & 1) == 0) return {};
			else return { 0, 1 };
		}
		vector<UL> ans;
		if ((B & (1 << (N - 1))) == 0) {
			vector<UL> buf1 = Solve2(N - 1, B);
			UL L = buf1.size(); if (!L) return {};
			vector<UL> buf2 = Solve2(N - 1, buf1[1]);
			ans.push_back(buf1[0]);
			rep(i, buf2.size()) ans.push_back(buf2[i] ^ (1 << (N - 1)));
			rep(i, L - 1) ans.push_back(buf1[i + 1]);
		}
		else {
			vector<UL> buf1 = Solve2(N - 1, 1);
			UL f = buf1[1];
			vector<UL> buf2 = Solve2(N - 1, B ^ (1 << (N - 1)) ^ 1);
			UL L = buf2.size(); if (!L) return {};
			rep(i, L) buf2[i] |= (1 << (N - 1));
			rep(i, L) buf2[i] ^= 1;
			rep(i, L) buf1.push_back(buf2[i]);
			ans = move(buf1);
		}
		return move(ans);
	}

	void Solve() {
		UL N, A, B; cin >> N >> A >> B;
		vector<UL> ans = Solve2(N, A^B);
		bool ok = true;
		if (ans.size() == 0) { cout << "NO" << endl; return; }

		rep(i, ans.size()) ans[i] ^= A;
		cout << "YES" << endl;
		rep(i, ans.size()) { if (i) cout << " "; cout << ans[i]; }
		cout << endl;
	}


	Problem();
};
int main() {
	unique_ptr<Problem> p(new Problem());
	p->Solve();
	return 0;
}
Problem::Problem() {
	cout << fixed << setprecision(10);
}