
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 $f(x) = x^{2} + a\ln x$.  
$(Ⅰ)$ When $a=-2$, find the extreme value of the function $f(x)$;  
$(Ⅱ)$ If $g(x) = f(x) + \frac{2}{x}$ is a monotonically increasing function on the interval $\left[1,+\infty \right)$, find the range of real numbers for $a$.

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