
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 that the center of the hyperbola $E$ is at the origin, and $P(3,0)$ is a focus of $E$. A line $l$ passing through $P$ intersects $E$ at points $A$ and $B$, and the midpoint of $AB$ is $N(-12,-15)$. Find the equation of the hyperbola $E$.

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