
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
Biejia and Vasha are playing a game. Biejia selects 100 non-negative numbers \(x_1, x_2, \cdots, x_{100}\) (they can be the same), whose sum equals 1. Vasha then pairs these numbers into 50 pairs in any way he chooses, computes the product of the two numbers in each pair, and writes the largest product on the blackboard. Biejia 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 be the number written on the blackboard under optimal strategy?

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