
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 $E$ with the equation $\frac{x^2}{t} + \frac{y^2}{3} = 1$, its foci are on the x-axis, $A$ is the left vertex of $E$, and a line with slope $k(k > 0)$ intersects $E$ at points $A$ and $M$. Point $N$ is on $E$, and $MA \perp NA$.  
(i) When $t=4$ and $|AM| = |AN|$, find the area of $\triangle AMN$;  
(ii) When $2|AM| = |AN|$, determine the range of values for $k$.

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