
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
Let set A be $[0, \frac{1}{2})$ and B be $[\frac{1}{2}, 1]$. Define the function $f(x)$ as follows:
$$
f(x) = 
\begin{cases} 
x + \frac{1}{2}, & x \in A \\
2(1 - x), & x \in B
\end{cases}
$$
If $x_0 \in A$ and $f[f(x_0)] \in A$, then the range of $x_0$ is __________________.

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