
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 $f(x)=|x|$.<br/>$(1)$ Solve the inequality $f(x)+f(x-1)\leqslant 2$;<br/>$(2)$ For $a+b=1\ (a,b>0)$ and $\forall x\in R$, the inequality $f(x-m)-|x+2|\leq\frac{1}{a}+\frac{1}{b}$ always holds. Find the range of real number $m$.

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