Problem: 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.

<image>

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:

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

<image>

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.

Solution (in CPP):

Solution:
#include <cstring>
#include <iostream>
#include <algorithm>
#include <bitset>
#include <map>
#include <vector>

using namespace std;

const int N = 1e5 + 10, M = 5e5 + 10, INF = 0x3f3f3f3f;
typedef long long ll;

int n, m, t, k, x;
int s[N], g[N];
int fa[N];
int deep[N];
int maxx;
int num[N];
int cnt;
int lson[N];
int head[N], ver[M], edge[M], nex[M], tot;
bitset<N> f[2007];
int val[N];
map<int, int> vis;
bool ok[N];
vector<int> v[N];

void dfs1(int x)
{
	deep[x] = deep[fa[x]] + 1;
	num[deep[x]]++;
	maxx = max(maxx, deep[x]);
	if (head[x] == -1)
	    lson[deep[x]]++;
	for (int i = head[x]; ~i; i = nex[i])
	{
		int y = ver[i];
		dfs1(y);
	}
}

void dfs2(int x, int t)
{
	if (!x) return;
	for (int i = 0; i < (int) v[x].size(); i++)
	{
		if (val[x] > t 
		|| f[x-1][t])
		    break;
		t -= val[x], ok[v[x][i]] = true;
	}
	dfs2(x-1, t);
}

void add(int x, int y)
{
	ver[tot] = y;
	nex[tot] = head[x];
	head[x] = tot++;
}

void init()
{
	memset(head, -1, sizeof head);
	tot = 0;
}

void solve()
{
	init();
	cin >> n >> x;
	for (int i = 2; i <= n; i++)
	{
		cin >> fa[i];
		add(fa[i], i);
	}
	dfs1(1);
	for (int i = 1; i <= maxx; i++)
	{
		if (vis[num[i]]) v[vis[num[i]]].push_back(i);
		else
		{
			vis[num[i]] = ++cnt;
			val[cnt] = num[i];
			v[cnt].push_back(i);
		}
	}
	f[0][0] = 1;
	for (int i = 1; i <= cnt; i++)
	{
		f[i] = f[i-1];
		int Size = v[i].size();
		for (int j = 1; j <= Size; j <<= 1)
		{
			Size -= j;
			f[i] |= f[i] << (j * val[i]);
		}
		f[i] |= f[i] << (Size * val[i]);
	}
	
	if (f[cnt][x])
	{
		printf("%d\n", maxx);
		dfs2(cnt, x);
		for (int i = 1; i <= n; i++)
		    putchar(ok[deep[i]]?'a':'b');
	}
	else
	{
		int res = INF;
		for (int i = x; i >= 0; i--)
		{
			if (f[cnt][i])
			{
				res = i;
				break;
			}
		}
		dfs2(cnt, res);
		int pos = -1;
		for (int i = 1; i <= maxx; i++)
		{
			if (!ok[i] && lson[i] >= x - res)
			{
				pos = i;
				break;
			}
		}
		printf("%d\n", maxx + 1);
		for (int i = 1; i <= n; i++)
		{
			if (deep[i] == pos && head[i] == -1)
				if (res == x)
				    putchar('b');
				else
				    putchar('a'), res++;
			else
			    putchar(ok[deep[i]]?'a':'b');
		}
	}
	return;
}

int main()
{
	solve();
	return 0;
}