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


- $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$ ;
- for any integers  $a \ge b \ge 0$ , we have  $g(a+b) \le f(a) + f(b)$ .

Determine the maximum possible value of  $g(0) + g(1) + g(2) + \cdots + g(84)$  over all such pairs of functions.

*Evan Chen (adapting from TST3, by Sean Li)*

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