
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 a quadratic equation in terms of $x$: $x^{2}-(2m+3)x+m^{2}+2=0$.  
$(1)$ If the equation has real roots, find the range of real numbers for $m$.  
$(2)$ If the two roots of the equation are $x_{1}$ and $x_{2}$, and satisfy ${x}_{1}^{2}+{x}_{2}^{2}=3x_{1}x_{2}-14$, find the value of the real number $m$.

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