
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 $f(x)=|x-1|$, where $x\in\mathbb{R}$.

$(1)$ Find the solution set of the inequality $|f(x)-3|\leqslant 4$;

$(2)$ If $f(x)+f(x+3)\geqslant m^2-2m$ always holds, find the range of the real number $m$.

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