
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
There are 800 points on a circle labeled clockwise as $1, 2, \cdots, 800$. They divide the circumference into 800 gaps. One point is selected and painted red. Then, the remaining points are painted red one by one according to the following rule: If the $k$-th point is painted red, then moving clockwise by $k$ gaps, the point reached is also painted red. What is the maximum number of red points that can be obtained on the circle? Provide a proof for your conclusion.

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