
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 the set $M=\{x|1 \lt x \lt 4\}$ and the set $N=\{x|3 \lt x \lt 5\}$.  
$(1)$ Find $M\cap N$ and $M\cup (\complement _{R}N)$;  
$(2)$ Let $A=\{x|a\leqslant x\leqslant a+3\}$. If $A\cup (\complement _{R}N)=R$, find the range of real number $a$.

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