
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 $f\left(x\right)=|x-1|-|x+2|$.  
$(1)$ Find the solution set of the inequality $f\left(x\right)\leqslant 1$;  
$(2)$ If the maximum value of $f\left(x\right)$ is $m$, and positive numbers $a$ and $b$ satisfy $a+b=m$, find the minimum value of $\frac{3}{a}+\frac{a}{b}$.

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