
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\dfrac{1}{2}$, $A$ and $B$ are respectively the left and right vertices of $C$, and $F$ is the left focus of $C$. Point $P$ is a moving point on $C$, and the maximum area of $\triangle ABP$ is $2\sqrt{3}$.

$(1)$ Find the equation of $C$;

$(2)$ A line $l$ passes through $F$, and $l$ intersects $C$ at two points $M$ and $N$. Is there a fixed point $D(m,0)$ such that $\overrightarrow{DM} \cdot \overrightarrow{DN}$ is a constant? If so, find $m$; if not, explain why.

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