
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
When simplifying the fraction $\frac{2x}{{x}^{2}-1}-\frac{1}{x-1}$, student A's solution is as follows. Read student A's solution and complete the following questions.  
Solution: Original expression $=\frac{2x}{(x+1)(x-1)}-\frac{1}{x-1}\ldots \ldots $①  
$=\frac{2x}{(x+1)(x-1)}•\left(x+1\right)\left(x-1\right)-\frac{1}{x-1}\cdot \left(x+1\right)\left(x-1\right)\ldots \ldots $②  
$=2x-\left(x+1\right)\ldots \ldots $③  
$=2x-x-1\ldots \ldots $④  
$=x-1.\ldots \ldots $⑤  
$(1)$ Student A made a mistake starting from step ____(fill in the number);  
$(2)$ Please write down the correct solution.

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