
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
Consider pairs  $(f,g)$  of functions from the set of nonnegative integers to itself such that 


- $f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
- $f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
- for any 20 nonnegative integers  $n_1, n_2, \dots, n_{20}$ , not necessarily distinct, we have   $$ g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}). $$

Determine the maximum possible value of  $g(0)+g(1)+\dots+g(6000)$  over all such pairs of functions.

*Sean Li*

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