
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
Let \( ABC \) be an isosceles right triangle with a right angle at vertex \( C \). The triangle is moved in the plane such that vertices \( A \) and \( B \) always lie on the perpendicular lines \( a \) and \( b \), respectively. (The orientation of the \( ABC \) triangle is fixed.) What path does the vertex \( C \) trace during this movement?

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