
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
The polygons $\mathcal{K}$ and $\mathcal{L}$ are positioned on the plane in such a way that their vertices are not on the extended lines of the other polygon's sides. At most, how many points do their sides intersect, if $\mathcal{K}$ is a pentagon and
a) $\mathcal{L}$ is a triangle;
b) $\mathcal{L}$ is a quadrilateral?

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