
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) defined as:
$$
f(x) =
\begin{cases}
x + \frac{1}{2}, & x \in A \\
2(1 - x), & x \in B
\end{cases}
$$
1. Find the value of $f[f(\frac{1}{3})]$
2. If $x_0 \in A$ and $f[f(x_0)] \in A$, find the range of possible values for $x_0$.

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