
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 function $f(x)= \dfrac {\ln (x^{2}-2x+a)}{x-1}$.  
$(1)$ When $a=1$, discuss the monotonicity of $f(x)$ on $(1,+\infty)$.  
$(2)$ If the domain of $f(x)$ is $(-\infty,1)\cup(1,+\infty)$.  
  $(i)$ Find the range of the real number $a$.  
  $(ii)$ If the inequality $f(x) < (x-1)\cdot e^{x}$ holds for any $x\in(1,+\infty)$, find the range of the real number $a$.

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