
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 $f\left(x\right)=|x-1|-|x+m|\left(m\in R\right)$.<br/>$(1)$ When $m=2$, find the solution to the inequality $f\left(x\right)+2 \lt 0$;<br/>$(2)$ If the inequality $f\left(x\right)+|x-4| \gt 0$ holds for all $x\in \left[0,2\right]$, find the range of real number $m$.

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