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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
#include <set>
#include <map>
#include <queue>
#include <random>
#include <chrono>

using namespace std;

typedef long long ll;
typedef long double ld;
#define fastInp cin.tie(0); cout.tie(0); ios_base::sync_with_stdio(0);

#define int long long

signed main()
{
    fastInp;
    ll n, a, b;
    cin >> n >> a >> b;
    ll dist = 0;
    for (int i = 0; i < n; i++) {
        if (((a >> i) & 1) != ((b >> i) & 1)) dist++;
    }

    if (dist != 1) {
        cout << "NO";
    } else {
        cout << "YES\n";
        ll ind = -1;
        vector<ll> ans;

        for (int i = 0; i < n; i++) {
            if (((a >> i) & 1) != ((b >> i) & 1)) {
                ind = i;
            } else {
                ll bt = (1ll << i), nbt = 0;
                if (!((a >> i) & 1)) swap(bt, nbt);

                if (ans.size() == 0) {
                    ans.push_back(bt);
                    ans.push_back(nbt);
                } else {
                    vector<ll> cop = ans;
                    reverse(cop.begin(), cop.end());
                    for (auto cur : cop) ans.push_back(cur);
                    for (int i2 = 0; i2 < cop.size(); i2++) ans[i2] |= bt;
                    for (int j = cop.size(); j < ans.size(); j++) ans[j] |= nbt;
                }

            }
        }

        ll bt = (1ll << ind), nbt = 0;
        if (!((a >> ind) & 1)) swap(bt, nbt);

        if (ans.size() == 0) {
            ans.push_back(bt);
            ans.push_back(nbt);
        } else {
            vector<ll> cop = ans;
            reverse(cop.begin(), cop.end());
            for (auto cur : cop) ans.push_back(cur);
            for (int i = 0; i < cop.size(); i++) ans[i] |= bt;
            for (int j = cop.size(); j < ans.size(); j++) ans[j] |= nbt;
        }

        for (auto cur : ans) cout << cur << " ";
    }

    return 0;
}