
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{1}{2}$, and its right focus is $F$, the minimum distance from a point on ellipse $E$ to point $F$ is 2.
(1) Find the equation of ellipse $E$;
(2) Let the left and right vertices of ellipse $E$ be $A$ and $B$, respectively. A line $l$ passing through point $A$ intersects ellipse $E$ and the line $x=8$ at points $M$ and $N$, respectively.
   (ⅰ) When the radius of the circle passing through points $A$, $F$, and $N$ is minimized, find the equation of this circle;
   (ⅱ) If $\cos\angle AMB= -\frac{\sqrt{65}}{65}$, find the area of $\triangle ABM$.

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