
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 equations of the two asymptotes of a hyperbola are $y= \pm \sqrt {2}x$, and the hyperbola passes through the point $(3, -2 \sqrt {3})$.  
(1) Find the equation of the hyperbola;  
(2) The left and right foci of the hyperbola are denoted as $F_1$ and $F_2$, respectively, and $P$ is a point on the hyperbola such that $\angle F_1PF_2 = 60°$. Find the area of $\triangle PF_1F_2$.

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