
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
If the two foci of the hyperbola $\Gamma_{1}$ are the two vertices of the ellipse $\Gamma_{2}: \frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1$, and the two directrices of the hyperbola $\Gamma_{1}$ pass through the two foci of the ellipse $\Gamma_{2}$, then the equation of the hyperbola $\Gamma_{1}$ is $\quad$ .

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