
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 the Cartesian coordinate plane $(xOy)$, there is an ellipse $(E)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{1}{2}$ and a right focus $F$. The minimum distance from a point on the ellipse $(E)$ to point $F$ is $2$.
$(1)$ Find the values of $a$ and $b$;
$(2)$ 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.
$\quad\quad①$ Find the equation of the circle with the minimum radius that passes through points $A$, $F$, and $N$;
$\quad\quad②$ 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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