
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 a hyperbola centered at the origin with its foci on the $x$-axis, a line passing through the right focus of the hyperbola with a slope of $\sqrt{\frac{3}{5}}$ intersects the hyperbola at two points $P$ and $Q$. If $O P \perp O Q$ (where $O$ is the origin) and $|P Q|=4$, find the equation of the hyperbola.


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