
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
A hyperbola $E$ passes through point $A(4,6)$, with its axes of symmetry being the coordinate axes, and its foci $F_1$ and $F_2$ are on the x-axis, with an eccentricity $e = 2$.

(I) Find the equation of the hyperbola $E$;  
(II) Find the equation of the line on which the angle bisector of $\angle F_1AF_2$ lies.

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