
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 circle with 800 points labeled in sequence clockwise as \(1, 2, \ldots, 800\), dividing the circle into 800 arcs. Initially, one point is painted red, and subsequently, additional points are painted red according to the following rule: if the \(k\)-th point is already red, the next point to be painted red is found by moving clockwise \(k\) arcs from \(k\). What is the maximum number of red points that can be obtained on the circle? Explain the reasoning.

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