
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 the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.

$(1)$ Find the coordinates of the foci $F_1$ and $F_2$; and calculate the distance from the focus $F_2$ to the asymptote;

$(2)$ If $P$ is a point on the hyperbola and $\angle F_1PF_2 = 30^\circ$, find the area $S$ of $\triangle F_1PF_2$.

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