
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 function $f(x) = \frac{k + x}{k - x} \cdot e^{x}$ $(k \in \mathbb{R})$.
- (I) If $k = 1$, find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
- (II) Determine the intervals of monotonicity for the function $f(x)$;
- (III) Suppose $k \leqslant 0$, if the function $f(x)$ has an extremum point in the interval $(\sqrt{3}, 2\sqrt{2})$, find the range of $k$.

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