
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 $x$: $x^{2}+\left(2m-1\right)x+m^{2}=0$.  
$(1)$ If the equation has real roots, find the range of real numbers for $m$.  
$(2)$ If $x_{1}$ and $x_{2}$ satisfy $x_{1}x_{2}+x_{1}+x_{2}=4$, find the value of $m$.

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