
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 $x=1$, $y=1$, and for all $x\in \left(2,5\right)$, $y \gt 0$ always holds, find the range of real number $a$;<br/>$(2)$ If $x=1$, $y=1$, and for all $a\in \left[-2,-1\right]$, $y \gt 0$ always holds, find the range of real number $x$.

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