
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 centered at the origin with its left focus F($-\sqrt{3}$, 0) and right vertex A(2, 0).

(1) Find the standard equation of ellipse C;  
(2) A line l with a slope of $\frac{1}{2}$ intersects ellipse C at points A and B. Find the maximum value of the chord length |AB| and the equation of line l at this time.

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