
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
Given a triangle touching side BC at point C', consider the lengths of the 12 segments which are heights of triangles A'B'C', A'_aB'_aC'_a, A'_bB'_bC'_b, and A'_cB'_cC'_c.

a) What is the maximum number of different lengths among these segments?

b) Find all possible numbers of different lengths.

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