
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1(-3, 0)$ and $F\_2(3, 0)$, respectively. The line $y = kx$ intersects the ellipse at points $A$ and $B$.

(I) If the perimeter of triangle $AF\_1F\_2$ is $4\sqrt{3} + 6$, find the standard equation of the ellipse.

(II) If $|k| > \frac{\sqrt{2}}{4}$, and a circle with diameter $AB$ passes through the right focus of the ellipse, find the range of the eccentricity $e$ of the ellipse.

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