
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
Given two sequences \(\{a_n\}\) and \(\{b_n\}\) that satisfy
\[
\begin{array}{l}
a_1 = 2, \ b_1 = 1, \\
a_{n+1} = 5a_n + 3b_n + 7 \ \text{for any positive integer} \ n, \\
b_{n+1} = 3a_n + 5b_n \ \text{for any positive integer} \ n,
\end{array}
\]
derive the general term formula for \(\{a_n\}\), \(a_n\).

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