
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, $f(x)=\sqrt{|x+1|+|x-3|-m}$, its domain is $\mathbb{R}$.

$(1)$ Find the range of the real number $m$;

$(2)$ If the maximum value of the real number $m$ is $n$, and the positive real numbers $a$, $b$ satisfy $\frac{2}{3a+b}+\frac{1}{a+2b}=n$, find the minimum value of $7a+4b$.

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