
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
Selected Optional 4-4: Coordinate Systems and Parametric Equations

In the polar coordinate system, the ray $l$: $θ = \frac{π}{6}$ intersects with the circle $C$: $ρ = 2$ at point $A$. The equation of the ellipse $Γ$ is: $ρ^2 = \frac{3}{1 + 2\sin^2 θ}$. Establish a rectangular coordinate system $xOy$ with the pole as the origin and the polar axis as the positive semi-axis of $x$.

(I) Find the rectangular coordinates of point $A$ and the parametric equation of ellipse $Γ$;

(II) If $E$ is the lower vertex of ellipse $Γ$ and $F$ is any point on ellipse $Γ$, find the maximum value of $\overrightarrow{AE} \cdot \overrightarrow{AF}$.

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