
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 \(\{b_n\}\) satisfying \(a_0 = 1\), \(b_0 = 0\), and the recurrence relations
\[
\begin{cases}
a_{n+1} = 7a_n + 6b_n - 3, \\
b_{n+1} = 8a_n + 7b_n - 4,
\end{cases}
\]
for \(n = 0, 1, 2, \ldots\), find \(a_n\).

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