
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 $f\left(x\right)=\sqrt{|{x+2|+|x-4}|-m}$ with the domain of $R$.  
$(1)$ Find the range of real number $m$;  
$(2)$ If the maximum value of $m$ is $n$, and for positive numbers $a$ and $b$ satisfying $\frac{4}{{a+5b}}+\frac{1}{{3a+2b}}=n$, find the minimum value of $4a+7b$.

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