
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
Let  $S=\{1,2,\dots,3000\}$ . Determine the maximum possible integer  $X$  that satisfies the condition:
For all bijective function  $f:S\rightarrow S$ , there exists bijective function  $g:S\rightarrow S$  such that  $$ \displaystyle\sum_{k=1}^{3000}\left(\max\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}-\min\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}\right)\geq X $$  

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