
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
We drew the incircle and the excircles of a triangle, as well as the six lines that touch two of these circles each, but are distinct from the sides of the triangle. How many ways can one choose from these six lines:

a) the side lines of a trapezoid,

b) the side lines of a parallelogram, assuming that none of the sides of the original triangle are equal in length?

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