
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
An isosceles triangle \( \triangle ABC \) has a base angle of \( \alpha \) and a leg length of \( a \). For every inscribed triangle within \( \triangle ABC \) (where an inscribed triangle is defined as having at least one vertex on each side of \( \triangle ABC \)), determine the minimum value of the longest side of these inscribed triangles. Also, describe the shape of this inscribed triangle when this minimum value is achieved.

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