
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]


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Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, $F$ is the right focus of $C$. A line $l$ passing through point $F$ and not coinciding with the $x$-axis intersects $C$ at points $A$ and $B$. When $l$ is parallel to the $y$-axis, $|AB|=3$.  
$(1)$ Find the equation of $C$;  
$(2)$ Let $P$ be the left vertex of $C$. Lines $PA$ and $PB$ intersect the line $x=4$ at points $D$ and $E$, respectively. Find the value of $\overrightarrow{FD}•\overrightarrow{FE}$.

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