
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 polynomial sequence  $(p_{n}(x))$  satisfying  $p_{1}(x)=x^{2}-1$ ,  $p_{2}(x)=2x(x^{2}-1)$ , and  $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$ , for  $n\geq 2$ , let  $s_{n}$  be the sum of the absolute values of the coefficients of  $p_{n}(x)$ . For each  $n$ , find a non-negative integer  $k_{n}$  such that  $2^{-k_{n}}s_n$  is odd.

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