
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 \( P_{n}(x) = \sum_{1 \leqslant 2k+1 \leq n} C_{n}^{2k+1} x^{n-2k-1}(x^{2}-1)^{k-1} \), let \( S_{n} \) be the sum of the absolute values of all the coefficients of the polynomial \( P_{n}(x) \). For any positive integer \( n \), find the exponent of 2 in the prime factorization of \( S_{n} \).

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