
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 $A=\{x| m+1 \leqslant x \leqslant 3m-1\}$ and $B=\{x| 1 \leqslant x \leqslant 10\}$.  
$(1)$ If $m=5$, find the intersection of sets $A$ and $B$.  
$(2)$ If $A \subseteq B$, find the range of real numbers for $m$.

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