
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
Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the product of the numbers at the endpoints is allocated to each segment. Then, the sum of the numbers on the red segments is subtracted by the sum of the numbers on the blue segments. What is the maximum possible result?

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