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: Runtime Error)

#include <bits/stdc++.h>
using namespace std;
using ll = long long;

#define ALL(obj) (obj).begin(),(obj).end()
#define SPEED cin.tie(0);ios::sync_with_stdio(false);

template<class T> using PQ = priority_queue<T>;
template<class T> using PQR = priority_queue<T,vector<T>,greater<T>>;

constexpr long long MOD = (long long)1e9 + 7;
constexpr long long MOD2 = 998244353;
constexpr long long HIGHINF = (long long)1e18;
constexpr long long LOWINF = (long long)1e15;
constexpr long double PI = 3.1415926535897932384626433L;

template <class T> vector<T> multivector(size_t N,T init){return vector<T>(N,init);}
template <class... T> auto multivector(size_t N,T... t){return vector<decltype(multivector(t...))>(N,multivector(t...));}
template <class T> void corner(bool flg, T hoge) {if (flg) {cout << hoge << endl; exit(0);}}
template <class T, class U>ostream &operator<<(ostream &o, const map<T, U>&obj) {o << "{"; for (auto &x : obj) o << " {" << x.first << " : " << x.second << "}" << ","; o << " }"; return o;}
template <class T>ostream &operator<<(ostream &o, const set<T>&obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr) o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
template <class T>ostream &operator<<(ostream &o, const multiset<T>&obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr) o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
template <class T>ostream &operator<<(ostream &o, const vector<T>&obj) {o << "{"; for (int i = 0; i < (int)obj.size(); ++i)o << (i > 0 ? ", " : "") << obj[i]; o << "}"; return o;}
template <class T, class U>ostream &operator<<(ostream &o, const pair<T, U>&obj) {o << "{" << obj.first << ", " << obj.second << "}"; return o;}
void print(void) {cout << endl;}
template <class Head> void print(Head&& head) {cout << head;print();}
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) {cout << head << " ";print(forward<Tail>(tail)...);}
template <class T> void chmax(T& a, const T b){a=max(a,b);}
template <class T> void chmin(T& a, const T b){a=min(a,b);}
std::vector<std::string> split(const std::string &str, const char delemiter) {std::vector<std::string> res;std::stringstream ss(str);std::string buffer; while( std::getline(ss, buffer, delemiter) ) res.push_back(buffer); return res;}
void YN(bool flg) {cout << (flg ? "YES" : "NO") << endl;}
void Yn(bool flg) {cout << (flg ? "Yes" : "No") << endl;}
void yn(bool flg) {cout << (flg ? "yes" : "no") << endl;}

int main() {
    int N,A,B,C; cin >> N >> A >> B;
    C=A^B;
    corner(!(__builtin_popcount(C)%2),"NO");
    vector<vector<int>> vv(N);
    vv[0]={0,1};
    for(int i=1;i<N;++i) {
        int M=vv[i-1].size();
        for(int j=0;j<M;++j) vv[i].push_back(2*vv[i-1][j]);
        for(int j=M-1;0<=j;--j) vv[i].push_back(2*vv[i-1][j]+1);
    }
    vector<int> bf(N),af;
    for(int i = 0; i < N; ++i) bf[i]=((C>>i)&1);
    af = bf;
    sort(ALL(af));
    vector<int> idx(N,-1);
    for(int i=0; i<N; ++i) {
        for(int j=0;j<N;++j) {
            if(idx[j]!=-1)continue;
            if(af[i]!=bf[j]) continue;
            idx[j]=i;
            break;
        }
    }
    int pre=0;
    vector<int> cnt;
    for(int i=N-1;1<=i;--i) {
        if(!af[i]) break;
        for(int x:vv[i-1]) cnt.push_back(pre|x);
        // print(i,bitset<10>(pre),vv[i-1]);

        pre ^= (1<<i);

        reverse(ALL(vv[i-1]));
        // print(i,bitset<10>(pre),vv[i-1]);
        for(int x:vv[i-1]) cnt.push_back(pre|x);
    }
    vector<int> ans;
    for(int& x:cnt) {
        int y = 0;
        for(int i = 0; i < N; ++i) {
            int j=idx[i];
            int b=((x>>j)&1);
            y ^= (b<<i);
        }
        ans.push_back(y);
    }
    for(int& x:ans) {
        x ^= A;
    }
    assert(ans.front() == A);
    assert(ans.back() == B);
    cout << "YES" << endl;
    for(int x:ans) cout << x << " ";
    cout << endl;
    return 0;
}