
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
Define \(\varphi^{k}(n)\) as the number of positive integers that are less than or equal to \(n / k\) and relatively prime to \(n\). Find \(\varphi^{2001}\left(2002^{2} - 1\right)\). (Hint: \(\varphi(2003) = 2002\).)

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