
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
In the plane, there are two polygons $P_1$ and $P_2$. $P_1$ is a regular $n_1$-sided polygon, and $P_2$ is an irregular $n_2$-sided polygon, where $n_1\le n_2$. Assuming $P_1$ and $P_2$ overlap but share no complete side, find the maximum number of points at which their boundaries can intersect.
A) $2n_1$
B) $2n_2$
C) $n_1n_2$
D) $n_1 + n_2$
E) None of these

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