
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
On a plane, given two mutually perpendicular lines \( l_{1} \) and \( l_{2} \) and a circle \( \mathcal{S} \), construct a right triangle \( ABC \) with a given acute angle, where vertices \( A \) and \( B \) lie on the lines \( l_{1} \) and \( l_{2} \), respectively, and the vertex \( C \), the right angle vertex, lies on the circle \( \mathcal{S} \).

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