
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 inequality in $x$, ${x}^{2}+2mx+m+2\geqslant 0$, has the solution set $R$.

$(1)$ Find the range of real numbers $m$.

$(2)$ Find the minimum value of the function $f(m)=m+ \dfrac{3}{m+2}$.

$(3)$ Solve the quadratic inequality in $x$, ${x}^{2}+(m-3)x-3m > 0$.

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