
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
The center of the hyperbola is the origin O, its imaginary axis is $2 \sqrt {6}$, the right focus is F(c, 0) (c > 0), and the line l: $x= \frac {a^{2}}{c}$ intersects the x-axis at point A, with |OF| = 3|OA|. A line passing through point F intersects the hyperbola at points P and Q.  
(Ⅰ) Find the equation of the hyperbola;  
(Ⅱ) If $\overrightarrow {AP} \cdot \overrightarrow {AQ} = 0$, find the equation of line PQ.

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