
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-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and at the segments, the products of the numbers at the endpoints were placed. Then, the sum of the numbers on the red segments was subtracted by the sum of the numbers on the blue segments. What is the largest possible value that could be obtained?

I. Bogdanov

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