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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