
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, right vertex $A$, and upper vertex $B$. $\triangle BF\_1F\_2$ is an equilateral triangle with a side length of $2$.
(I) Find the standard equation and eccentricity of the ellipse $C$;
(II) Determine if there exists a line $l$ passing through the focus $F\_2$, intersecting the ellipse at two points $P$ and $Q$, such that $PA$ is parallel to $QF\_1$. If it exists, find the equation of the line $l$; if not, explain the reason.

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