
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 nonnegative integers  $p$ ,  $q$ ,  $r$ , let \[
f(p, q, r) = (p!)^p (q!)^q (r!)^r.
\]Compute the smallest positive integer  $n$  such that for any triples  $(a,b,c)$  and  $(x,y,z)$  of nonnegative integers satisfying  $a+b+c = 2020$  and  $x+y+z = n$ ,  $f(x,y,z)$  is divisible by  $f(a,b,c)$ . 

*Proposed by Brandon Wang*

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