
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 function $f\left(x\right)=x^{2}+2\left(a-1\right)x+2a+6$.<br/>$(Ⅰ)$ If the function $f\left(x\right)$ is monotonically increasing in $\left[4,+\infty \right)$, find the range of real number $a$;<br/>$(Ⅱ)$ If the inequality $f\left(x\right)\geqslant 0$ holds for any $x\in R$, find the range of real number $a$;<br/>$(Ⅲ)$ If the equation $f\left(x\right)=0$ has two distinct real roots greater than $1$, find the range of real number $a$.

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