
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 $l: \theta= \dfrac{\pi}{6}$ intersects with the circle $C: \rho=2$ at point $A$. The equation of the ellipse $\Gamma$ is given by $\rho^2= \dfrac {3}{1+2\sin^2\theta}$. By setting the origin as the pole and the polar axis as the positive $x$-axis, establish the Cartesian coordinate system $xOy$.
(Ⅰ) Find the Cartesian coordinates of point $A$ and the parametric equations of the ellipse $\Gamma$;
(Ⅱ) If $E$ is the lower vertex of the ellipse $\Gamma$, and $F$ is any point on ellipse $\Gamma$, find the range of the scalar product $ \overrightarrow{AE} \cdot \overrightarrow{AF}$.

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