
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 that the domain of the function $f\_M(x)$ is the set of real numbers $R$, and it satisfies $f\_M(x)= \begin{cases} 1,x\in M \\ 0,x\notin M\end{cases}$ ($M$ is a non-empty proper subset of $R$), there are two non-empty proper subsets $A$, $B$ in $R$, and $A\cap B=\emptyset$, then the range of $F(x)= \dfrac {f\_{A\cup B}(x)+1}{f\_A(x)+f\_B(x)+2}$ is ___.

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