
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
In a polar coordinate system, the ray $l: \theta = \frac{\pi}{6}$ intersects the circle $C: \rho = 2$ at point $A$, and the equation of the ellipse $\Gamma$ is $\rho^2 = \frac{3}{1+2\sin^2\theta }$. Set up a Cartesian coordinate system with the pole as the origin and the polar axis as the positive x-axis.

(Ⅰ) Find the Cartesian coordinates of point $A$ and the parametric equations of the ellipse $\Gamma$.
(Ⅱ) Let $E$ be the lower vertex of the ellipse $\Gamma$ and $F$ be any point on the ellipse $\Gamma$. Find the maximum value of the dot product $\overrightarrow{AE} \cdot \overrightarrow{AF}$.

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