
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]


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Elective 4-5 Inequality Lecture

Given that the solution set of the inequality $|x+3|-2x-1 < 0$ is $O$.

$(1)$ Find the value of $x_0$;

$(2)$ If the function $f(x)=|x-m|+|x+\frac{1}{m}|-x_0$ $(m > 0)$ has a zero, find the value of the real number $m$.

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