
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 positive integer \( n \geq 2 \), Borya and Vasha play a game. Borya selects \( 2n \) non-negative numbers (which can be the same) \( x_1, x_2, \ldots, x_{2n} \) such that their sum equals 1. Vasha then arranges these numbers in a circle, after which he calculates the product of each pair of adjacent numbers and writes the largest of these \( 2n \) products on the blackboard. Borya wants the number written on the blackboard to be as large as possible, while Vasha wants it to be as small as possible. What will the number written on the blackboard be, given the optimal strategies of both players?

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