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Problem Setting

The task in problem setting C is almost equivalent to problem setting A. Hence, for details on the task the contestant is refered to problem setting A. The only difference in Problem C is the shape of the higher order pseudo-Boolean function which serves as an input. This input will now have degree at most 6 ($d=6$), while coefficients are provided for all possible terms of the function. For details of the input problem contestants are referred to the section below.

Input Format

The following requirements shall be imposed on the input problem.

• $3 \\leq N \\leq 10$
• $1 \\leq K \\leq 2^N$
• $0 \\leq d_i \\leq 6$
• $1 \\leq \\left|c_i \\right| \\leq 100$
• $1 \\leq v_{i, j} \\leq N$
• For all $j$ $(1 \\leq j \\lt d_i)$ we require $v_{i, j} < v_{i, j+1}$
• For two lines $i,j$ with $i \\neq j$ we require $\\left\[ v_{i, 1}, \\dots , v_{i, d_i} \\right\] \\neq \\left\[ v_{j, 1}, \\dots, v_{j, d_j} \\right\]$, i.e., two separate lines shall not encode the same type of monomial.
• The line $d_i = 0$ should not exceed beyond the coefficient.
• All of the variables $k$ $(1 \\leq k \\leq N)$ should appear in at least one of the term, i.e., there exists at least one $v_{i,j}$ such that $k = v_{i, j}$.