
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 $C$ with one focus at $(0, \sqrt{3})$, and it passes through the point $P\left(\frac{1}{2}, \sqrt{3}\right)$.  
(1) Find the standard equation of ellipse $C$;  
(2) Given $A(1,0)$, line $l$ intersects ellipse $C$ at points $M$ and $N$, and $AM \perp AN$;  
(i) If $|AM| = |AN|$, find the equation of line $l$;  
(ii) If $AH \perp MN$ at $H$, find the equation of the trajectory of point $H$.

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