Zero xor

Input

The first line contains two integers $n$ and $m$ (1 \le n \le 10^6, 1 \le m \le 30) — the number of vertices and the bit limit of the values.

The second line contains $n$ integers a_1, a_2, \dots, a_n (0 \le a_i < 2^m) — the values associated with each vertex.

The next n-1 lines describe the edges of the tree. Each line contains two integers $u$ and $v$ (1 \le u, v \le n), meaning there is an edge between vertices $u$ and $v$.

Output

If no such subset exists, print NO (case-insensitive, e.g., No).

If a valid independent subset with XOR sum equal to $0$ exists, print YES in the first line (case-insensitive, e.g., yEs, YeS). In the second line, print a single integer k (1 \le k \le n) — the number of vertices in the subset. In the third line, print $k$ space-separated integers v_1, v_2, \dots, v_k, representing the indices of the selected vertices in any order.

Examples

Note

The tree for the first sample

Scoring

1. ($10$ points): a_i is a power of two for all 1 \le i \le n;

2. ($31$ points): n > 60;

3. ($31$ points): m \le 10;

4. ($53$ points): tree is a bamboo, that is, the edges are i \ i+1 for 1 \le i < n;

5. ($20$ points): tree is a star, that is, the edges are $1i$ for 2 \le i \le n;

6. ($45$ points): m \le 20;

7. ($85$ points): no additional constraints;