
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
Let the function $f(x)=|ax+1|+|x-a|$ $(a > 0)$, $g(x)=x^2+x$.

$(1)$ When $a=1$, find the solution set of the inequality $g(x) \geqslant f(x)$;

$(2)$ Given $f(x) \geqslant \frac{3}{2}$, find the range of values for $a$.

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