
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 $y=ax^{2}+bx+2$ where $a$ and $b$ are real numbers.  
$(1)$ If $x=1$, $y=1$, and for all $x\in (2,5)$, $y>0$ always holds, find the range of real number $a$.  
$(2)$ If $x=1$, $y=1$, and for all $a\in [-2,-1]$, $y>0$ always holds, find the range of real number $x$.  
$(3)$ For all $x\in \mathbb{R}$, when $b>0$, $y\geqslant 0$ always holds, find the minimum value of $\frac{a+2}{b}$.

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