
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 $t=ax^{2}-(a+2)x+2$.  
$(1)$ If $t \lt 3-2x$ holds for all $x\in R$, find the range of real number $a$;  
$(2)$ For any $a\in [-1,1]$, $t \gt 0$ holds, find the range of $x$;  
$(3)$ If there exists $m \gt 0$ such that the equation $ax^2-(a+2)|x|+2=m+\frac{1}{m}+1$ has four distinct real roots, find the range of real number $a$.

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