
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
Consider the sequence  $a_1, a_2, a_3, ...$  defined by  $a_1 = 9$  and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for  $n \ge 1$ .

Find all natural numbers  $n$  for which  $a_n$  is a perfect square of an integer.

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