
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
Given a natural number \( n \geq 2 \), Petya and Vasya play the following game: Petya selects \( 2n \) (not necessarily distinct) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) whose sum equals 1. Vasya arranges these numbers in a circle in any order he chooses. After this, he calculates the products of pairs of neighboring numbers and writes the largest of these \( 2n \) products on the board. Petya aims to maximize this number on the board, while Vasya aims to minimize it. What number will be on the board with optimal play?

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