
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
Let  $q=p^r$  for a prime number  $p$  and positive integer  $r$ . Let  $\zeta = e^{\frac{2\pi i}{q}}$ . Find the least positive integer  $n$  such that 
\[\sum_{\substack{1\leq k\leq q \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}\]
is not an integer. (The sum is over all  $1\leq k\leq q$  with  $p$  not dividing  $k$ .)

*Victor Wang*

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