
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 any positive integer  $n$ , and  $i=1,2$ , let  $f_{i}(n)$  denote the number of divisors of  $n$  of the form  $3 k+i$  (including  $1$  and  $n$  ). Define, for any positive integer  $n$ ,   $$ f(n)=f_{1}(n)-f_{2}(n) $$   Find the value of  $f\left(5^{2022}\right)$  and  $f\left(21^{2022}\right)$ .

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