
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
Find the maximum positive integer  $k$  such that for any positive integers  $m,n$  such that  $m^3+n^3>(m+n)^2$ , we have  $$ m^3+n^3\geq (m+n)^2+k $$  *Proposed by Dorlir Ahmeti, Albania*

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