
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 function $y=ax^{2}+bx+2$ ($a$ and $b$ are real numbers).<br/>$(1)$ If the function passes through the point $(1,1)$ and $y \gt 0$ for all $x\in R$, find the range of real number $a$; <br/>$(2)$ If the function passes through the point $(1,1)$ and $y \gt 0$ for all $a\in \left[-2,-1\right]$, find the range of real number $x$; <br/>$(3)$ For all $x\in R$, when $b \gt 0$, $y\geqslant 0$ always holds, find the minimum value of $\frac{{a+2}}{b}$.

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