
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 a positive integer  $n$ , let  $\Omega(n)$  denote the number of prime factors of  $n$ , counting multiplicity. Let  $f_1(n)$  and  $f_3(n)$  denote the sum of positive divisors  $d|n$  where  $\Omega(d)\equiv 1\pmod 4$  and  $\Omega(d)\equiv 3\pmod 4$ , respectively. For example,  $f_1(72) = 72 + 2 + 3 = 77$  and  $f_3(72) = 8+12+18 = 38$ . Determine  $f_3(6^{2020}) - f_1(6^{2020})$ . 

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