Problem E. Expression with period ≡

Statement

An expression for a function is composed of single-digit numbers, variable x, signs  +  and  * , and zero or one calls to function sin. More formally:

• expression ::= 0|1|2|3|4|5|6|7|8|9|x
• expression ::= expression+expression
• expression ::= expression*expression
• expression ::= sin(expression)

Function f(x) has period p if f(x + p) = f(x) for all x.

Your program must, given an expression with no more than one call to sin, determine the minimal period of a function.

Input file format

The input file contains a single string — expression.

Output file format

Output file must contain single a floating point number — period with relative error not greater than 10^− 6. If a function is constant or aperiodic, output number 0.

Constraints

Expression length is between 1 and 25 characters. There are no spaces in expression.

Sample tests

sin(x)+x

0

5*6*sin(2*x+5+x*7+x)+1

0.62831853

sin(x*0)

0