
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
Petya and Vasya are playing the following game. Petya chooses 100 (not necessarily distinct) non-negative numbers \( x_{1}, x_{2}, \ldots, x_{100} \) whose sum is equal to 1. Vasya divides them into 50 pairs as he wishes, calculates the product of the numbers in each pair, and writes the largest of the 50 products on the board. Petya wants the number on the board to be as large as possible, while Vasya wants it to be as small as possible. What number will be on the board with optimal play?

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