Curiosity Has No Limits

When Masha came to math classes today, she saw two integer sequences of length n - 1 on the blackboard. Let's denote the elements of the first sequence as a_i (0 ≤ a_i ≤ 3), and the elements of the second sequence as b_i (0 ≤ b_i ≤ 3).

Masha became interested if or not there is an integer sequence of length n, which elements we will denote as t_i (0 ≤ t_i ≤ 3), so that for every i (1 ≤ i ≤ n - 1) the following is true:

• a_i = t_i | t_{i + 1} (where | denotes the bitwise OR operation) and
• b_i = t_i & t_{i + 1} (where & denotes the bitwise AND operation).

The question appeared to be too difficult for Masha, so now she asked you to check whether such a sequence t_i of length n exists. If it exists, find such a sequence. If there are multiple such sequences, find any of them.

Input

The first line contains a single integer n (2 ≤ n ≤ 10^5) — the length of the sequence t_i.

The second line contains n - 1 integers a_1, a_2, …, a_{n-1} (0 ≤ a_i ≤ 3) — the first sequence on the blackboard.

The third line contains n - 1 integers b_1, b_2, …, b_{n-1} (0 ≤ b_i ≤ 3) — the second sequence on the blackboard.

Output

In the first line print "YES" (without quotes), if there is a sequence t_i that satisfies the conditions from the statements, and "NO" (without quotes), if there is no such sequence.

If there is such a sequence, on the second line print n integers t_1, t_2, …, t_n (0 ≤ t_i ≤ 3) — the sequence that satisfies the statements conditions.

If there are multiple answers, print any of them.

Examples

Input

4 3 3 2 1 2 0

Output

YES 1 3 2 0

Input

3 1 3 3 2

Output

NO

Note

In the first example it's easy to see that the sequence from output satisfies the given conditions:

• t_1 | t_2 = (01_2) | (11_2) = (11_2) = 3 = a_1 and t_1 & t_2 = (01_2) & (11_2) = (01_2) = 1 = b_1;
• t_2 | t_3 = (11_2) | (10_2) = (11_2) = 3 = a_2 and t_2 & t_3 = (11_2) & (10_2) = (10_2) = 2 = b_2;
• t_3 | t_4 = (10_2) | (00_2) = (10_2) = 2 = a_3 and t_3 & t_4 = (10_2) & (00_2) = (00_2) = 0 = b_3.

In the second example there is no such sequence.

inputFormat

Input

The first line contains a single integer n (2 ≤ n ≤ 10^5) — the length of the sequence t_i.

The second line contains n - 1 integers a_1, a_2, …, a_{n-1} (0 ≤ a_i ≤ 3) — the first sequence on the blackboard.

The third line contains n - 1 integers b_1, b_2, …, b_{n-1} (0 ≤ b_i ≤ 3) — the second sequence on the blackboard.

outputFormat

Output

In the first line print "YES" (without quotes), if there is a sequence t_i that satisfies the conditions from the statements, and "NO" (without quotes), if there is no such sequence.

If there is such a sequence, on the second line print n integers t_1, t_2, …, t_n (0 ≤ t_i ≤ 3) — the sequence that satisfies the statements conditions.

If there are multiple answers, print any of them.

Examples

Input

4 3 3 2 1 2 0

Output

YES 1 3 2 0

Input

3 1 3 3 2

Output

NO

Note

In the first example it's easy to see that the sequence from output satisfies the given conditions:

• t_1 | t_2 = (01_2) | (11_2) = (11_2) = 3 = a_1 and t_1 & t_2 = (01_2) & (11_2) = (01_2) = 1 = b_1;
• t_2 | t_3 = (11_2) | (10_2) = (11_2) = 3 = a_2 and t_2 & t_3 = (11_2) & (10_2) = (10_2) = 2 = b_2;
• t_3 | t_4 = (10_2) | (00_2) = (10_2) = 2 = a_3 and t_3 & t_4 = (10_2) & (00_2) = (00_2) = 0 = b_3.

In the second example there is no such sequence.

样例

4
3 3 2
1 2 0

YES
1 3 2 0 

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