
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, $F\_1$, $F\_2$ are its left and right foci, respectively, and $A$ is a point on the ellipse. The dot product of $\overrightarrow{AF_{2}}$ and $\overrightarrow{F_{1}F_{2}}$ is 0, and the slope of the line $AF_{1}$ is $\frac{\sqrt{3}}{12}$. The length of the major axis is 8.
(1) Find the equation of the ellipse $C$;
(2) The line $y=kx+ \frac{3}{2}(k \neq 0)$ intersects the ellipse $C$ at different points $E$ and $F$, and both $E$ and $F$ are on the circle with center $B(0,-2)$. Find the value of $k$.

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