
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 a sequence \(\left\{a_{n}\right\}\) that satisfies \(a_{0} = 3\), \(a_{1} = 9\), and \(a_{n} = 4a_{n-1} - 3a_{n-2} - 4n + 2\) for \(n \geqslant 2\), find all non-negative integers \(n\) such that \(a_n\) is divisible by 9.

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