
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 $f\left(x\right)=|2x+1|-|x-2|$.<br/>$(1)$ Find the solution set for the inequality $f\left(x\right)\leqslant 8$;<br/>$(2)$ Let the maximum value of the function $g\left(x\right)=f\left(x\right)-|x-2|$ be $m$. If positive numbers $a$ and $b$ satisfy $a+b=m$, find the minimum value of $\frac{1}{a}+\frac{9}{b}$.

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