
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
The standard equation of the hyperbola that shares foci with the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$ and passes through the point $P(2,\sqrt{2})$ is $\frac{{x}^{2}}{a^{2}}-\frac{{y}^{2}}{b^{2}}=1$, where $a$ and $b$ are positive real numbers. Find the equation of the hyperbola.

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