
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 a constant $a\neq 0$, $f(x)=a\ln x+2x$.
$(1)$ When $a=-4$, find the extremum of $f(x)$;
$(2)$ When the minimum value of $f(x)$ is not less than $-a$, determine the range of values for the real number $a$.

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