
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 at $F(-\sqrt{3}, 0)$ and right vertex at $A(2, 0)$.
(1) Find the standard equation of the ellipse $C$.
(2) A line $l$ with slope $\frac{1}{2}$ intersects ellipse $C$ at points $A$ and $B$. Find the maximum length of the chord $|AB|$ and the equation of line $l$ in this case.

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