
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
Regular polygons with $4, 5, 7,$ and $9$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
**(\textbf{A}) \: 52 \qquad (\textbf{B}) \: 56 \qquad (\textbf{C}) \: 58 \qquad (\textbf{D}) \: 62 \qquad (\textbf{E}) \: 66**

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