
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
Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning.

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