AB Tree

给定一棵 $n$ 个节点的树，根为 $1$ ，每个节点会分配到一个字符 `a` 或 `b`。 要求整棵树中字符 `a` 的数量为 $x$ ，字符 `b` 的数量为 $n-x$ 。 定义节点 $v$ 上的字符串： - 若 $v$ 是根节点，则 $v$ 的上的字符串为根节点分配到的字符。 - 否则，$v$ 上的字符串为父节点上的字符串的末尾加上 $v$ 分配到的字符。 请为每个节点分配字符，在满足字符 `a` ，`b` 数量要求的前提下，使得所有节点上的字符串的种类最少。 注：本翻译基于 [**fanfansann**](https://www.luogu.com.cn/user/262605) 的翻译修改。

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.

The first line contains two integers $ n $ and $ x $ ( $ 1 \leq n \leq 10^5 $ ; $ 0 \leq x \leq n $ ) — the number of vertices in the tree the number of a's. The second line contains $ n - 1 $ integers $ p_2, p_3, \dots, p_{n} $ ( $ 1 \leq p_i \leq n $ ; $ p_i \neq i $ ), where $ p_i $ is the parent of node $ i $ . It is guaranteed that the input describes a connected tree.

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.

9 3
1 2 2 4 4 4 3 1

4
aabbbbbba

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 $ \{\text{a}, \text{aa}, \text{aab}, \text{aabb}\} $ , so the number of distinct strings is $ 4 $ .