
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. The polygon with $4$ sides shares one vertex with the polygon with $9$ sides, and no other vertices are shared among the polygons. No three sides of these polygons intersect at a common point inside the circle. How many points inside the circle do two of their sides intersect?
A) 400
B) 420
C) 452
D) 480
E) 500

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