
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
The Devil and the Man play a game. Initially, the Man pays some cash  $s$  to the Devil. Then he lists some  $97$  triples  $\{i,j,k\}$  consisting of positive integers not exceeding  $100$ . After that, the Devil draws some convex polygon  $A_1A_2...A_{100}$  with area  $100$  and pays to the Man, the sum of areas of all triangles  $A_iA_jA_k$ . Determine the maximal value of  $s$  which guarantees that the Man receives at least as much cash as he paid.

*Proposed by Nikolai Beluhov, Bulgaria*

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