# Related Recurrence Relations

Algebra Level 5

Let $a_n$, $b_n$, and $c_n$ satisfy the system of recursive relations below for $n \geq 2$:

$\large \begin{cases} b_n - 2^{n-2} = c_{n-1} - c_{n-2} - 3a_{n-2} \\ 2^{n-3} + a_{n-1} - 5a_{n-2} = 2b_{n-1} + c_{n-2} \\ c_n - 6a_{n-2} + 5a_{n-1} = a_n - 2^{n-2} \end{cases}$

If $a_0 = 5$, $a_1 = 10$, $b_0 = -\frac{259}{48}$, $b_1 = -\frac{65}{8}$, $c_0 = \frac{7}{4}$, and $c_1 = \frac{3}{2}$, find the value of $a_7 + b_5c_6$.

This problem is part of the set "Xenophobia"

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