Flipped parentheses P64777

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

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

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.