
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 the two foci of the hyperbola \( C \) be exactly the two vertices of the ellipse \( \Gamma \): \(\frac{x^2}{16}+\frac{y^2}{9}=1 \), and let the two vertices of the hyperbola \( C \) be exactly the two foci of the ellipse \( \Gamma \). Then the equation of the hyperbola \( C \) is _______.

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