
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 the right focus $F\_2$ of the ellipse $C\_1: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ coincides with the focus of the parabola $C\_2: y^{2}=8x$, and the left endpoint is $(-\sqrt{6},0)$.
(1) Find the equation of the ellipse;
(2) The line $l\_2$ passing through the right focus of ellipse $C\_1$ with a slope of $\sqrt{3}$ intersects the ellipse $C\_1$ at points A and B. Find the length of the chord AB.

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