
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  $\varphi(n)$  denotes the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ .  Determine the number of positive integers  $2\leq n\leq 50$  such that all coefficients of the polynomial
\[
\left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1\leq k\leq n\gcd(k,n) = 1}}(x-k)
\]
are divisible by  $n$ .

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