
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
Point $P$ is on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, with $F_1$ and $F_2$ being the left and right foci, respectively. If $\angle F_1PF_2 = 60^\circ$,

$(1)$ Find the area of $\triangle F_1PF_2$;  
$(2)$ Find the coordinates of point $P$.

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