
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 $f(x)=2|x-2|+ax$ $(x \in \mathbb{R})$ has a minimum value.

    $(1)$ Find the range of values for the real number $a$;

    $(2)$ Let $g(x)$ be an odd function defined on $\mathbb{R}$, and for $x < 0$, $g(x)=f(x)$. Find the expression for $g(x)$.

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