
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
a) Each vertex at the base of a triangle is connected by straight lines to \( n \) points located on the opposite side. Into how many parts do these lines divide the triangle?

b) Each of the three vertices of a triangle is connected by straight lines to \( n \) points located on the opposite side of the triangle. Into how many parts do these lines divide the triangle, assuming that no three of them intersect at the same point?

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