
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 that $F\_1$ is the left focus of the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, and the line $l$ passing through $F\_1$ intersects the ellipse at two points $P$ and $Q$.

(I) If the slope angle of line $l$ is $45^{\circ}$, find the length of segment $|PQ|$.
(II) Assume the slope of line $l$ is $k (k \neq 0)$, the point symmetric to $P$ about the origin is $P'$, and the point symmetric to $Q$ about the $x$-axis is $Q'$. The slope of line $P'Q'$ is $k'$. If $|k'| = 2$, find the value of $k$.

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