
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
Let  $ABC$  be a triangle. Take  $n$  point lying on the side  $AB$  (different from  $A$  and  $B$ ) and connect all of them by straight lines to the vertex  $C$ . Similarly, take  $n$  points on the side  $AC$  and connect them to  $B$ . Into how many regions is the triangle  $ABC$  partitioned by these lines?

Further, take  $n$  points on the side  $BC$  also and join them with  $A$ . Assume that no three straight lines meet at a point other than  $A,B$  and  $C$ . Into how many regions is the triangle  $ABC$  partitioned now?

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