Maximum Valid Parentheses Depth

Given a string that represents a mathematical expression, determine the maximum depth of valid nested parentheses. The expression contains only digits, the operators +, -, *, /, and parentheses ( ). If the input string contains any invalid characters (i.e. characters other than 0-9, +, -, *, /, (, )) or if the parentheses are not properly balanced, output -1.

For example, for the expression \( (1+(2*3)+((8)/4))+1 \), the maximum depth of valid nested parentheses is 3.

Note: All formulas should be written in \( \LaTeX \) format when applicable.

inputFormat

The input consists of a single line containing the expression string.

outputFormat

Output a single integer: the maximum depth of valid nested parentheses if the expression is valid, or -1 if it contains any invalid characters or unbalanced parentheses.

## sample

(1+(2*3)+((8)/4))+1

3

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