Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\]  Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]


Let's think step by step and output the final answer within \boxed{}.
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