You are given a tree T with n vertices (numbered 1 through n) and a letter in each vertex. The tree is rooted at vertex 1.

Let's look at the subtree T_v of some vertex v. It is possible to read a string along each simple path starting at v and ending at some vertex in T_v (possibly v itself). Let's denote the number of distinct strings which can be read this way as $\text{[math]}$ .

Also, there's a number c_v assigned to each vertex v. We are interested in vertices with the maximum value of $\text{[math]}$ .

You should compute two statistics: the maximum value of $\text{[math]}$ and the number of vertices v with the maximum $\text{[math]}$ .

Input

The first line of the input contains one integer n (1 ≤ n ≤ 300 000) — the number of vertices of the tree.

The second line contains n space-separated integers c_i (0 ≤ c_i ≤ 10⁹).

The third line contains a string s consisting of n lowercase English letters — the i-th character of this string is the letter in vertex i.

The following n - 1 lines describe the tree T. Each of them contains two space-separated integers u and v (1 ≤ u, v ≤ n) indicating an edge between vertices u and v.

It's guaranteed that the input will describe a tree.

Output

Print two lines.

On the first line, print $\text{[math]}$ over all 1 ≤ i ≤ n.

On the second line, print the number of vertices v for which $\text{[math]}$ .

Note

In the first sample, the tree looks like this:

$\text{[math]}$

The sets of strings that can be read from individual vertices are:

$\text{[math]}$

Finally, the values of $\text{[math]}$ are:

$\text{[math]}$

In the second sample, the values of $\text{[math]}$ are (5, 4, 2, 1, 1, 1). The distinct strings read in T₂ are $\text{[math]}$ ; note that $\text{[math]}$ can be read down to vertices 3 or 4.

Acyclic Organic Compounds

Examples