0112. Polynomial Equation

Input file name:	equation.in
Output file name:	equation.out
Time limit:	2 min
Memory limit:	64 megabytes

You are given some polynomials P₀, P₁, …, P_m on four variables X, Y, Z and T. Your task is to test whether for all sets a=(x,y,z,t)∈C⁴ such that P₁(x,y,z,t)=…=P_m(x,y,z,t)=0 the equality P₀(x,y,z,t)=0 also holds.

Input file

The first line of the input file contains integer number m, 1 ≤ m ≤ 10. The rest of the input file contains polynomials P₀, P₁, …, P_m, separated by dot characters `.'. The total degree of each polynomial will not exceed 10, and all coefficients are integer numbers not exceeding 10⁵ by their absolute values. The exact representation of polynomials is given by the following BNF:

Different tokens of input (with the exception of digits forming a <number>) are separated by any number of space and/or newline characters.

Output file

In the single line of output print either YES or NO (without quotes).

Examples:

equation.in	equation.out
2 X^2 + 3*X*Y - 2Y^2 . X^4 . X^2 + Y^2 .	YES

Source: Petrozavodsk Summer 2003. Primary Contest, Sunday, August 24
Author: Andrew Lopatin, Nick Durov

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