
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
Selected Exercise $(4-5)$: Inequality Lecture

Given that the domain of the function $f(x)=\sqrt{|x+2|+|x-4|-m}$ is $\mathbb{R}$.

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

$(II)$ If the maximum value of $m$ is $n$, and the positive numbers $a$ and $b$ satisfy $\frac{4}{a+5b}+\frac{1}{3a+2b}=n$, find the minimum value of $4a+7b$.

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