
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
For the function $y=f(x)$, $x\in R$ and given real numbers $a$, $b$ ($a \lt b$), if there exists a positive real number $t$ such that the function $y=f(x)$ is either increasing or decreasing on both intervals $[a,b]$ and $[a+t,b+t]$, then the function $y=f(x)$, $x\in R$ is called an $\Omega(t)$ function on the interval $[a,b]$;  
$(1)$ Given $a=0$, $b=1$, please indicate whether the functions $y_{1}=\sin x$ and $y_{2}=\cos x$ are $\Omega(2)$ functions on the interval $[0,1]$ (no need to provide reasoning);  
$(2)$ Given $a=-\frac{π}{2}$, $b=\frac{π}{2}$, and the function $y=\sin x$ is an $\Omega(t)$ function on the interval $[a,b]$, please write down all possible values of $t$ and provide reasoning;  
$(3)$ If the function $y=\sin x$ is both an $\Omega(\beta)$ function on the interval $[a,\alpha +B]$ and an $\Omega(a)$ function on the interval $[B,\alpha +\beta]$, find the range of $\sin \alpha +\sin \beta$ as $\alpha$ and $\beta$ vary over all possible values.

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