
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 its left and right vertices respectively as $A$ and $B$, and its eccentricity $e = \dfrac{1}{2}$, let point $P$ be a variable point on the ellipse. The maximum area of $\triangle PAB$ is $2\sqrt{3}$.
1. Find the standard equation of the ellipse.
2. A variable line $l$ passes through the left focus $F_1$ of the ellipse $C$ and intersects the ellipse at points $M$ and $N$. Is there a fixed point $D$ on the x-axis such that $\overrightarrow{DM} \cdot \overrightarrow{DN}$ is constant? If such a point exists, find the coordinates of $D$ and the constant value. If no such point exists, please provide the explanation.

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