
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 $f(x)=|x+m|+2|x-1|$ where $(m > 0)$.

$(1)$ When $m=1$, find the solution set of the inequality $f(x) \leqslant 10$;

$(2)$ If the solution set of the inequality $f(x+1) \geqslant 3$ is $\mathbb{R}$, find the range of the real number $m$.

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