
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 the sequence \(\{a_n\}\) defined by \(a_1 = 1\), \(a_2 = 3\), and for all \(n \in \mathbf{N}\), \(a_{n+2} = (n+3)a_{n+1} - (n+2)a_n\), find all values of \(n\) for which \(a_n\) is divisible by 11.

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