
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 sets $A=\{x|x^{2}-3x-10 \lt 0\}$ and $B=\{x\left|\right.2-m\leqslant x\leqslant 2+m,m \gt 0\}$.  
$(1)$ If $m=4$, find $A$⋃$B$ and $(\complement _{R}A)$⋂$B$.  
$(2)$ If "$x\in A$" implies "$x\in B$" is true for ______, find the range of real number $m$.  
Choose one from "① sufficient but not necessary condition, ② necessary but not sufficient condition" and fill in the blank line above for answering. Note: If multiple conditions are chosen for answering separately, the first answer will be scored.

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