
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 $f(x)=|2x+2|-5$

$(1)$ Solve the inequality: $f(x)-|x-1|\geqslant 0$;

$(2)$ When $m > 0$, the graph of the function $g(x)=f(x)+|x-m|$ exactly encloses a triangle with the $x$-axis. Find the range of the real number $m$.

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