Flipped parentheses

A CME (Correct Mathematical Expression) was defined by the following rules:

z is a CME;
if $X$ is a CME, then so is ( $X$ );
if $X$ and $Y$ are both CMEs, then so is ( $X$ + $Y$ );
there are no more CMEs than those produced by the three rules above.

This set of rules produces terms like

(z) (z+z) (z+(z+z)) ((((z)))) …

Unfortunately, the job to produce the CMEs was given to a half-crazy computer (a HAL’s cousin) that sometimes flipped the parentheses, from ‘)’ to ‘(’ and viceversa, thus producing terms like

)z) (z+z( (z+(z+z() ))))z(((( …

We call these terms ACMEs (Almost Correct Mathematical Expressions). You are asked to write a program such that, given an ACME, computes the minimum number of parentheses that must be flipped to get a CME.

Input

The input has several non-empty strings consisting of at most $10^4$ characters chosen from {‘z’, ‘(’, ‘)’, ‘+’}.

Output

For each string of the input, tell if it is a CME, an ACME, or rubbish. In the second case, compute the minimum number of flips to convert the string into a CME.

Problem information

Author: Salvador Roura

Generation: 2026-01-25T11:21:26.755Z