
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 $C$ with foci $F\_1(-2,0)$, $F\_2(2,0)$, and passing through point $P(7,12)$.
(1) Find the equations of the hyperbola $C$ and its asymptotes;
(2) If a line $l$ with slope $1$ intersects the hyperbola $C$ at points $A$ and $B$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$ (where $O$ is the origin), find the equation of line $l$.

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