
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 $A=\{1,4,7,10\}$, $B=\{x| m < x < m+9\}$, $C=\{x| 3 \leq x \leq 6\}$.  
$(1)$ When $m=1$, find $A\cup B$ and $A \cap (\overline{C})$.  
$(2)$ If $B \cap C = C$, find the range of values for $m$.

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