
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 chooses non-negative numbers \(x_{1}, x_{2}, \ldots, x_{11}\) such that their sum is 1. Vasya arranges them in a row at his discretion, calculates the products of adjacent numbers, and writes on the board the largest of the resulting ten products. 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 the best play from both Petya and Vasya?

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