
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 $f(x)=|x+2a|+|x-1|$.

$(1)$ If $a=1$, solve the inequality $f(x) \leqslant 5$;

$(2)$ When $a \neq 0$, let $g(a)=f\left( \frac{1}{a} \right)$, find the range of values of $a$ that satisfy $g(a) \leqslant 4$.

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