
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 that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, respectively. Line $l$ passes through $F\_2$ and intersects the ellipse $C$ at points $A$ and $B$. The angle of inclination of line $l$ is $60^{\circ}$, and the distance from $F\_1$ to line $l$ is $2 \sqrt{3}$.
(I) Find the focal length of the ellipse $C$;
(II) If $\overrightarrow{AF_{2}}=2 \overrightarrow{F_{2}B}$, find the equation of the ellipse $C$.

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