
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
For  $p \in \mathbb{R}$ , let  $(a_n)_{n \ge 1}$  be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of  $p$  for which the series  $\sum_{n=1}^\infty a_n$  converges.


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