
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 the sets $A=[0, \frac{1}{2})$ and $B=[\frac{1}{2}, 1]$ and the function $f(x)= \begin{cases} x+ \frac{1}{2}, & x\in A \\ 2(1-x), & x\in B \end{cases}$, if $x_0 \in A$ and $f[f(x_0)] \in A$, determine the range of $x_0$.

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