
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
Given the ellipse \( x^{2} + a^{2} y^{2} = a^{2} \) (where \( a > 1 \)), one of its vertices \( C(0,1) \) is the right-angle vertex of an inscribed right triangle \( ABC \). What is the maximum number of such triangles that can be constructed?

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