Acyclic Organic Compounds
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 . 

Also, there's a number *c**v* assigned to each vertex *v*. We are interested in vertices with the maximum value of .

You should compute two statistics: the maximum value of  and the number of vertices *v* with the maximum .
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.
Print two lines. 

On the first line, print  over all 1<=≤<=*i*<=≤<=*n*. 

On the second line, print the number of vertices *v* for which .
In the first sample, the tree looks like this:

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

Finally, the values of <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/14691bbbb07694ceb84c7c73aa24a29ecfcb9e6b.png" style="max-width: 100.0%;max-height: 100.0%;"/> are:

In the second sample, the values of <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/58314b06b4c754bc3d1eeee1a5f0140ed5645015.png" style="max-width: 100.0%;max-height: 100.0%;"/> are (5, 4, 2, 1, 1, 1). The distinct strings read in *T*<sub class="lower-index">2</sub> are <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/072d38bb85cddb83248af087aadc065527254da5.png" style="max-width: 100.0%;max-height: 100.0%;"/>; note that <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/439f442d1a281ff76510cd09766962dff9fdbd65.png" style="max-width: 100.0%;max-height: 100.0%;"/> can be read down to vertices 3 or 4.