
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 $f(x)=|2x-4|+|x+1|$.<br/>$(1)$ Find the solution set of the inequality $f(x)\geqslant 4$;<br/>$(2)$ Let $g(x)=f(x)-|x-2|$. If the minimum value of $g(x)$ is $m$, where $a$, $b$, and $c$ are positive real numbers and $a+b+c=m$, find the minimum value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.

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