
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 $f(x) = x + \frac{a}{x}$, where $a \in \mathbb{R}$. The domain of the function $y = f(x)$ is $D$.  
$(1)$ When $a = 2$ and $D = (-\infty, 0)$, find the range of the function $y = f(x)$.  
$(2)$ When $a = -1$ and $D = (0, +\infty)$, determine the monotonicity of the function $y = f(x)$ and explain the reason.

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