
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
The sequence of polynomials  $\left( P_{n}(X)\right)_{n\in Z_{>0}}$  is defined as follows: $P_{1}(X)=2X$  $P_{2}(X)=2(X^2+1)$  $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$ , for all positive integers  $n$ .
Find all  $n$  for which  $X^2+1\mid P_{n}(X)$ 

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