
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
The function $f\_M(x)$ is defined on the domain $\mathbb{R}$ as follows: $$f\_M(x)= \begin{cases} 2,x\in M \\\\ 0,x\notin M\\end{cases}$$ where $M$ is a non-empty proper subset of the real number set $\mathbb{R}$. There are two non-empty proper subsets $A$, $B$ in $\mathbb{R}$ that satisfy $A\cap B=\emptyset$. The range of the function $F(x)=\frac{f\_A(x)+f\_B(x)+2}{f\_{A\cup B}(x)+2}$ is \_\_\_\_\_\_.

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