
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 $P=\{x|2x^{2}-3x+1\leqslant 0\}$ and $Q=\{x\left|\right.\left(x-a\right)\left(x-a-1\right)\leqslant 0\}$.  
$(1)$ Let the universal set $U=R$, if $a=1$, find $(\complement _{U}P)$⋂$Q$;  
$(2)$ If _____ (please choose from ① $P$⋂$Q=P$, ② $x\in P$ is a sufficient condition for $x\in Q$, ③ $P\cup Q=Q$), find the range of real number $a$.  
Note: If multiple conditions are chosen for answering separately, the first answer will be scored.

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