
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: $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) with eccentricity $\frac {\sqrt {6}}{3}$, and the distance from one end of the minor axis to the right focus is $\sqrt {3}$.

(1) Find the equation of ellipse C.

(2) Suppose a line $\ell$ with slope 1 passes through the left focus and intersects ellipse C at points A and B. Find the length of chord AB.

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