
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 $f\left(x\right)=|x-2|+|x-4|$.<br/>$(1)$ Solve the inequality with respect to $x$: $f\left(x\right)\geqslant 4$;<br/>$(2)$ If the minimum value of $f\left(x\right)$ is $M$, and positive numbers $a$ and $b$ satisfy $a+b=M$, find the minimum value of $\frac{1}{a}+\frac{{b+4}}{b}$.

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