Maximum Depth of a Tree

A tree's depth is how many nodes its longest path from the root down to a leaf has. Read a binary tree and print its depth. An empty tree has depth 0.

How a tree is written

A tree comes on one line, read level by level and left to right. After a node come two places for its children, left first, and a child that isn't there is written null. The nulls at the very end are left out, and an empty tree is just null.

        8
      /   \
     4     11
    /     /  \
   2     9    15

This tree is written 8 4 11 2 null 9 15: first 8, then its children 4 and 11, then the children of 4 (2 and none), then those of 11 (9 and 15). The children of 2, 9 and 15 would all be null at the end, so they are left out.

The starter code already reads the input and builds the tree out of Nodes with value, left and right, then prints what max_depth returns. You write max_depth.

Input

• Line 1: the tree, written as above. The tree has at most 500 nodes, each value between -10^4 and 10^4.

Output

One integer: the depth of the tree.

Examples

Input:
8 4 11 2 null 9 15

Output:
3

Input:
5 null 7 null 9

Output:
3

The tree is one branch going right: 5, then 7, then 9, so three nodes deep.

Input:
42

Output:
1

A lone root is one node deep.

Input:
null

Output:
0

An empty tree has no nodes at all.