
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 $E:$ $\frac{{x}^{2}}{t}+ \frac{{y}^{2}}{3}=1$ with foci on the $x$-axis, $A$ is the left vertex of $E$, and a line with slope $k(k > 0)$ intersects $E$ at points $M$ and $A$, and point $N$ is on $E$ with $MA \perp NA$.

(I) When $t=4$ and $|AM|=|AN|$, find the area of triangle $AMN$.

(II) When $2|AM|=|AN|$, find the range of $k$.

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