
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 the quadratic equation in $x$: $x^{2}-4x-2m+5=0$ has two real roots.<br/>$(1)$ Find the range of real numbers for $m$;<br/>$(2)$ If $x_{1}$ and $x_{2}$ are the two roots of the equation, and satisfy $x_{1}x_{2}+x_{1}+x_{2}=m^{2}+6$, find the value of $m$.

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