
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 $A=\{x| -3 \leqslant x \leqslant 6\}$ and $B=\{x| 6-m \lt x \lt m+3\}$.  
$(1)$ When $m=6$, find $(\complement _{R}A) \cup B$;  
$(2)$ If $A \cup B = A$, find the range of real numbers for $m$.

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