
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 $P=\{x|3a-10\leqslant x \lt 2a+1\}$ and $Q=\{x||2x-3|\leqslant 7\}$.  
$(1)$ When $a=2$, find $P\cap (\complement _{R}Q)$;  
$(2)$ If "$x\in P$" is a necessary but not sufficient condition for "$x\in Q$", find the range of real numbers $a$.

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