
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  $n$  be a positive integer and let  $a_1, a_2, \cdots a_k$  be all numbers less than  $n$  and coprime to  $n$  in increasing order. Find the set of values the function  $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$ .

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