
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
Point \( A \) is located between two parallel lines at distances \( a \) and \( b \) from them. This point serves as the vertex of an angle \( \alpha \), forming various possible triangles whose other two vertices lie on these lines. Find the minimum area value of such triangles.

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