
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 \( a_n = (n^2 + 1) 3^n \), find a recurrence relation \( a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0 \). Hence evaluate \( \sum_{n \geq 0} a_n x^n \).

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