
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 $f\left(x\right)=\sqrt{\left|x+1\right|+\left|x-3\right|-m}$ with a domain of $R$.

(1) Find the range of real number values for $m$.
(2) If the maximum value of $m$ is $n$, and positive numbers $a$ and $b$ 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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