
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
The one-variable quadratic equation about $x$ is $x^{2}-4x-2m+5=0$ with two real roots $x_{1}$ and $x_{2}$, and $x_{1}\neq x_{2}$.  
$(1)$ Find the range of real number $m$;  
$(2)$ Given that ${x}_{1}{x}_{2}+{x}_{1}+{x}_{2}={m}^{2}+6$, find the value of $m$.

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