
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 that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-2\left(m+1\right)x+m^{2}+3=0$.  
$(1)$ Find the range of values for $m$.  
$(2)$ Determine if there exists a real root $m$ such that $(x_{1}-1)(x_{2}-1)=m+6$ holds. If it exists, find the value of $m$; if not, explain the reason.

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