
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 it 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 $F_{1}$ and $F_{2}$, respectively. Point $P$ is on the hyperbola, and $\angle F_{1}PF_{2}$ is $60^{\circ}$. Find the area $S_{\triangle PF_{1}F_{2}}$.

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