
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 the ellipse $$C_{1}: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ whose right focus $F_{2}$ 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) A line $l_{2}$ passing through the right focus of the ellipse $C_{1}$ with a slope of $\sqrt{3}$ intersects the ellipse $C_{1}$ at chord AB. Find the length of this chord.

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