
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 $f(x) = x - \frac{1}{x} + a\ln{x}$, where $a \in \mathbb{R}$:
1. If the function $f(x)$ is monotonically increasing on the interval $[1, +\infty)$, determine the range of the real number $a$.
2. Let $g(x) = \frac{1}{2}x^2 + (m-1)x + \frac{1}{x}$, where $m \leq -\frac{3\sqrt{2}}{2}$, and $h(x) = f(x) + g(x)$. When $a=1$, $h(x)$ has two critical points $x_1$ and $x_2$, with $x_1 < x_2$. Find the minimum value of $h(x_1) - h(x_2)$.

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