
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where its left and right vertices are A and B, respectively, and its eccentricity $e = \frac{1}{2}$. Point P is a moving point on the ellipse, and the maximum area of $\triangle PAB$ is $2\sqrt{3}$.
(Ⅰ) Find the standard equation of the ellipse;
(Ⅱ) A moving 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 a constant? If it exists, find the coordinates of point D and the constant value; if not, explain why.

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