
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 $f(x)=x- \frac {1}{x}+a\ln x$ $(a\in\mathbb{R})$.

$(1)$ If the function $f(x)$ is monotonically increasing on $[1,+\infty)$, find the range of values for the real number $a$;

$(2)$ Given $g(x)= \frac {1}{2}x^{2}+(m-1)x+ \frac {1}{x}$, $(m\leqslant - \frac {3 \sqrt {2}}{2})$, $h(x)=f(x)+g(x)$, when $a=1$, $h(x)$ has two extreme points $x_{1}$, $x_{2}$, and $x_{1} < x_{2}$, find the minimum value of $h(x_{1})-h(x_{2})$.

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