
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 $f\left(x\right)=|x+2|+2|x-1|$.<br/>$(1)$ Solve the inequality $f\left(x\right)\leqslant 4$;<br/>$(2)$ Let the minimum value of the function $f\left(x\right)$ be $m$, $a+2b=m\left(a \gt 1,b \gt 0\right)$, find the minimum value of $\frac{1}{a-1}+\frac{2}{b}$.

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