
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
Determine the maximum number  $ h$  satisfying the following condition: for every  $ a\in [0,h]$  and every polynomial  $ P(x)$  of degree 99 such that  $ P(0)\equal{}P(1)\equal{}0$ , there exist  $ x_1,x_2\in [0,1]$  such that  $ P(x_1)\equal{}P(x_2)$  and  $ x_2\minus{}x_1\equal{}a$ .

*Proposed by F. Petrov, D. Rostovsky, A. Khrabrov*

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