
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 \( C_{0}: x^{2} + y^{2} = 1 \) and \( C_{1}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) where \( a > b > 0 \), find the condition that \( a \) and \( b \) must satisfy such that for any point \( P \) on \( C_{1} \), there exists a parallelogram with \( P \) as a vertex, that is tangential to \( C_{0} \) externally and is inscribed in \( C_{1} \). Provide proof for your conclusion.

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