
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 quadratic equation in terms of $x$: $x^{2}-6x+2m-1=0$ has two real roots $x_{1}$ and $x_{2}$.  
$(1)$ If $x_{1}=1$, find the values of $x_{2}$ and $m$.  
$(2)$ Does there exist a real number $m$ such that $(x_{1}-1)(x_{2}-1)=\frac{6}{{m-5}}$? If yes, find the value of $m$; if not, explain the reason.

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