
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) Given three different line segments, we need to construct a triangle $ABC$ in which the side $AB$ is equal to one of the segments, and the distances of the first and third quarter points of side $BC$ from point $A$ are equal to the other two segments, respectively.

b) Choose specific values for the three segments in some cases and determine, by choosing the roles of the segments in every possible way, how many cases yield a solution.

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