
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
Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $\frac{x^{2}}{4}+y^{2}=1$ respectively.  
$(1)$ If $M$ is a point on the ellipse and $\angle F_{1}MF_{2}=120^{\circ}$, find the area of $\triangle F_{1}MF_{2}$.  
$(2)$ If $P$ is a moving point on the ellipse, find the maximum and minimum values of $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}$.

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