
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For a positive integer  $n$ , we say an  $n$ -*transposition* is a bijection  $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$  such that there exist exactly two elements  $i$  of  $\{1,2, \dots , n\}$  such that  $\sigma(i) \neq i$ .
	Fix some four pairwise distinct  $n$ -transpositions  $\sigma_1,\sigma_2,\sigma_3,\sigma_4$ . Let  $q$  be any prime, and let  $\mathbb{F}_q$  be the integers modulo  $q$ . Consider all functions  $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$  that satisfy, for all integers  $i$  with  $1 \leq i \leq n$  and all  $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$ , \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all  $x_1,\ldots,x_n\in\mathbb{F}_q^n$  and all  $\sigma\in\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$ , \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\]
(Note that the equalities in the previous sentence are in  $\mathbb F_q$ . Note that, for any  $a_1,\ldots ,a_n, b_1, \ldots , b_n \in \mathbb{F}_q$ , we have  $(a_1,\ldots , a_n)+(b_1, \ldots, b_n)=(a_1+b_1,\ldots, a_n+b_n)$ , where  $a_1+b_1,\ldots , a_n+b_n \in \mathbb{F}_q$ .)
	For a given tuple  $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ , let  $g(x_1,\ldots,x_n)$  be the number of different values of  $f(x_1,\ldots,x_n)$  over all possible functions  $f$  satisfying the above conditions.
	Pick  $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$  uniformly at random, and let  $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)$  be the expected value of  $g(x_1,\ldots,x_n)$ . Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3,\sigma_4)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)-1}{q-1}\right)\right).\]
Pick four pairwise distinct  $n$ -transpositions  $\sigma_1,\sigma_2,\sigma_3,\sigma_4$  uniformly at random from the set of all  $n$ -transpositions. Let  $\pi(n)$  denote the expected value of  $\kappa(\sigma_1,\ldots,\sigma_4)$ . Suppose that  $p(x)$  and  $q(x)$  are polynomials with real coefficients such that  $q(-3) \neq 0$  and such that  $\pi(n)=\frac{p(n)}{q(n)}$  for infinitely many positive integers  $n$ . Compute  $\frac{p\left(-3\right)}{q\left(-3\right)}$ .

*Proposed by Gopal Goel*

Present the answer in LaTex format: \boxed{Your answer}
user