
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
Let the solution set of the inequality $\left(x-a\right)\left(x+1\right) \gt 0$ with respect to $x$ be $P$, and the solution set of the inequality $|x-1|\leqslant 1$ be $Q$.  
$(1)$ If $a=1$, find the set $(\complement _{R}P)\cup Q$;  
$(2)$ If $a \gt 0$ and $P\cap Q=\varnothing$, find the range of real numbers for $a$.

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