Parentheses Tree

A rooted ordered tree $T$ can be written as a string of matched parentheses $p(T)$. A tree with a single node is written as (). When the root of $T$ has $k$ ordered subtrees $T\_1, T\_2, \dots , T\_k$ attached as its children, $p(T)$ is defined as follows:

$p(T) ≔$ ( $+ p(T\_1) + p(T\_2) + \cdots + p(T\_k) +$ )

Here $+$ means concatenation of two strings. The figure below shows two rooted ordered trees. The string representations $p(T\_L)$ and $p(T\_R)$ are ((()()())()) and (()((()(()))())), respectively.

The distance from the root to a leaf is the number of edges on the path from the root to that leaf. In the figure, the root nodes are blue, and the distance from the root to each leaf is shown. The sums of the root-to-leaf distances are $7$ for $T\_L$ and $10$ for $T\_R$.

Given a string of matched parentheses that represents exactly one rooted ordered tree, output the sum of the distances from the root to all leaf nodes.

The input is one line containing a string of matched parentheses that represents one rooted ordered tree. The string contains no characters other than parentheses. Its length is at least $2$ and at most $10^7$.

Print one line containing the sum of the distances from the root of the tree to all leaf nodes.