AB Tree

Kilani and Abd are neighbors for 3000 years, but then the day came and Kilani decided to move to another house. As a farewell gift, Kilani is going to challenge Abd with a problem written by their other neighbor with the same name Abd.

The problem is:

You are given a connected tree rooted at node 1.

You should assign a character a or b to every node in the tree so that the total number of a's is equal to x and the total number of b's is equal to n - x.

Let's define a string for each node v of the tree as follows:

• if v is root then the string is just one character assigned to v:
• otherwise, let's take a string defined for the v's parent p_v and add to the end of it a character assigned to v.

You should assign every node a character in a way that minimizes the number of distinct strings among the strings of all nodes.

Input

The first line contains two integers n and x (1 ≤ n ≤ 10^5; 0 ≤ x ≤ n) — the number of vertices in the tree the number of a's.

The second line contains n - 1 integers p_2, p_3, ..., p_{n} (1 ≤ p_i ≤ n; p_i ≠ i), where p_i is the parent of node i.

It is guaranteed that the input describes a connected tree.

Output

In the first line, print the minimum possible total number of distinct strings.

In the second line, print n characters, where all characters are either a or b and the i-th character is the character assigned to the i-th node.

Make sure that the total number of a's is equal to x and the total number of b's is equal to n - x.

If there is more than one answer you can print any of them.

Example

Input

9 3 1 2 2 4 4 4 3 1

Output

4 aabbbbbba

Note

The tree from the sample is shown below:

The tree after assigning characters to every node (according to the output) is the following:

Strings for all nodes are the following:

• string of node 1 is: a
• string of node 2 is: aa
• string of node 3 is: aab
• string of node 4 is: aab
• string of node 5 is: aabb
• string of node 6 is: aabb
• string of node 7 is: aabb
• string of node 8 is: aabb
• string of node 9 is: aa

The set of unique strings is {a, aa, aab, aabb}, so the number of distinct strings is 4.

inputFormat

Input

The first line contains two integers n and x (1 ≤ n ≤ 10^5; 0 ≤ x ≤ n) — the number of vertices in the tree the number of a's.

The second line contains n - 1 integers p_2, p_3, ..., p_{n} (1 ≤ p_i ≤ n; p_i ≠ i), where p_i is the parent of node i.

It is guaranteed that the input describes a connected tree.

outputFormat

Output

In the first line, print the minimum possible total number of distinct strings.

In the second line, print n characters, where all characters are either a or b and the i-th character is the character assigned to the i-th node.

Make sure that the total number of a's is equal to x and the total number of b's is equal to n - x.

If there is more than one answer you can print any of them.

Example

Input

9 3 1 2 2 4 4 4 3 1

Output

4 aabbbbbba

Note

The tree from the sample is shown below:

The tree after assigning characters to every node (according to the output) is the following:

Strings for all nodes are the following:

• string of node 1 is: a
• string of node 2 is: aa
• string of node 3 is: aab
• string of node 4 is: aab
• string of node 5 is: aabb
• string of node 6 is: aabb
• string of node 7 is: aabb
• string of node 8 is: aabb
• string of node 9 is: aa

The set of unique strings is {a, aa, aab, aabb}, so the number of distinct strings is 4.

样例

9 3
1 2 2 4 4 4 3 1

4
aabbbbbba

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