## Problem E. Equation ≡

• problems
 Author: ACM ICPC 2007-2008, Northeastern European Regional Contest Time limit: 3 sec Input file: equation.in Memory limit: 64 Mb Output file: equation.out

### Statement

Your task is to solve an equation of the form f(x) = 0 where f(x) is written in postfix notation with numbers, operations  + ,  − ,  * ,  / , and at most one occurrence of a variable x. For example, f(x) for an equation (4x + 2) / 2 = 0 is written as: 4X * 2 + 2 /

The solution for f(x) = 0 is x =  − 1 / 2.

### Input file format

The input file consists of a single line with at most 30 tokens separated by spaces.

Each token is either:

• a digit from 0 to 9;
• an operation  + ,  − ,  * , or  / ;
• an uppercase letter X that denotes variable x.

The input file contains a correct representation of f(x) in postfix notation where token X occurs at most once. There is no division by a constant zero in this equation, that is, there always exists a value of x, such that f(x) can be evaluated without division by zero.

### Output file format

Write to the output file:

• X = p / q if equation f(x) = 0 has a single solution that can be represented with a simple fraction p / q, where p and q are coprime integer numbers and q is positive.
• NONE if equation f(x) = 0 has no solution;
• MULTIPLE if equation f(x) = 0 has multiple solutions.

### Sample tests

No. Input file (equation.in) Output file (equation.out)
1
4 X * 2 + 2 /
X = -1/2
2
2 2 *
NONE
3
0 2 X / *
MULTIPLE

0.086s 0.011s 13