
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 an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with left and right foci $F_{1}$, $F_{2}$, respectively, and an eccentricity of $\frac{1}{2}$. Let $M$ be a moving point on the ellipse, and the maximum area of $\triangle F_{1}MF_{2}$ is $\sqrt{3}$.

1. Find the equation of the ellipse.
2. Assume the left vertex of the ellipse is $A_{1}$, and a line $l$ passing through the right focus $F_{2}$ intersects the ellipse at points $A$ and $B$. Connect $A_{1}A$ and $A_{1}B$, and extend them to intersect the line $x=4$ at points $P$ and $Q$, respectively. Determine whether $\overrightarrow{PF_{2}} \cdot \overrightarrow{QF_{2}}$ is a constant value. If it is, find this constant value; otherwise, explain why not.

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