
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 $F_1(-2,0)$ and $F_2(2,0)$, which passes through the point $P(7,12)$,  
(1) Find the equation of the hyperbola C and the equations of its asymptotes;  
(2) Let a line $l$, with slope 1, intersect hyperbola C at points A and B, and let $\overrightarrow{OA} \perp \overrightarrow{OB}$ (where O is the origin of coordinates). Find the equation of line $l$.

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