
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 $a\in \mathbb{R}$, the function $f(x)=\left\{\begin{array}{l} \frac{3}{x}+a, \text{ for } x>0 \\ x^2+1, \text{ for } x\leq 0 \end{array}\right.$. If there exist three distinct real numbers $x_{1}$, $x_{2}$, $x_{3}$ such that $\frac{{f({x_1})}}{{{x_1}-1}}=\frac{{f({x_2})}}{{{x_2}-1}}=\frac{{f({x_3})}}{{{x_3}-1}}=-2$, then the range of real number $a$ is ______.

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