
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
If the one-variable quadratic equation in $x$ is $\left(m-1\right)x^{2}-2mx+m-2=0$ with two real roots $x_{1}$ and $x_{2}$.<br/>$(1)$ Determine the range of real numbers for $m$;<br/>$(2)$ If $(x_{1}+2)(x_{2}+2)-2x_{1}x_{2}=17$, find the value of $m$.

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