
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 \( n \geqslant 2 \), players A and B play a game. Player A selects \( 2n \) (possibly identical) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) whose sum equals 1. Player B then arranges them in some order on a circle. Subsequently, Player B calculates the product of each pair of adjacent numbers and writes the largest of these \( 2n \) products on the board. Player A wants the number written on the board to be as large as possible, while Player B wants it to be as small as possible. Question: What will the number written on the board be, assuming both players use optimal strategies?

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