Matrix nightmare

As Obi-Wan would put it: "This isn't the problem statement you are looking for. Move along."

The double factorial numbers are the numbers $d_i$ defined by the following recursive formula: $d_0 = 1$, and $\forall i > 0$: $d_i = d_{i-1} \cdot i!$. For example, $d_3 = 1! \cdot 2! \cdot 3! = 12$.

Sequences of length $n$ with all elements in the set $\{0, \dots ,n-1\}$ are called limited sequences of order $n$. For example, $(0, 2, 0, 1)$ is a limited sequence of order $4$. The set of all limited sequences of order $n$ is denoted $S_n$.

The spread factor of a sequence $A = (a_0, \dots , a_{n-1})$ is $\sigma(A) = \prod_{i=0}^{n-1}\prod_{j=i+1}^{n-1}(a_i - a_j)$.

There is a direct isomorphism between pairs of sequences and sequences of pairs. Let $A, B \in S_n$ with $A = (a_0, \dots , a_{n-1})$ and $B = (b_0, \dots , b_{n-1})$. The corresponding sequence of pairs $((a_0, b_0), \dots ,(a_{n-1}, b_{n-1}))$ is denoted $P_{A,B}$.

Pairs of integers are ordered lexicographically: $(a, b) \le (c, d)$ if either $a < c$, or ($a = c \land b \le d$). A sequence $P = (p_0, \dots , p_{n-1})$ of integer pairs is ordered lexicographically if $p_i \le p_{i+1}$ for all $i$. Let $\rho(P) =$ [if $P$ is ordered lexicographically then $1$ else $0$].

Let $M$ be an $n \times n$ matrix with rows and columns indexed from $0$ to $n - 1$. Its elements are $m_{r,c}$. A matrix is called a $\mathbb{Z}$-var matrix if each element is either an integer or a variable. The $n$-step traversal weight of $M$ is

$\varphi(M) = \frac{1}{d_{n-1}^2} \cdot \sum_{A=(a_0,\dots ,a_{n-1}), A \in S_n} \sum_{B=(b_0,\dots ,b_{n-1}), B \in S_n} \left( \rho(P_{A,B}) \cdot |\rho(A)| \cdot \rho(B) \cdot \prod_{i=0}^{n-1} m_{a_i,b_i} \right)$

Given is a multivariate polynomial $p$ with integer coefficients. Produce any reasonably small $\mathbb{Z}$-var matrix $M$ such that $\varphi(M) = p$.

The first line of the input contains an integer $t$, the number of test cases. Each test case is preceded by a blank line.

Each test case is a single line describing the polynomial. The variables are the letters a through z, and the syntax will be clear from the input file. Each polynomial has fewer than $50$ operations (additions, subtractions and multiplications).

For each test case, output one matrix in the following format: its size $n$, then all its elements in row major order. The elements may be separated by any positive amount of whitespace. The size of the matrix must not exceed $70$. All integers must be between $-10^9$ and $10^9$, inclusive.

If no such matrix exists for a given polynomial, output a single zero instead.