
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 \( p_{1}, p_{2}, p_{3}, \ldots \) be the prime numbers listed in increasing order, and let \( 0 < x_{0} < 1 \) be a real number between 0 and 1. For each positive integer \( k \), define

\[ x_{k}=\left\{
    \begin{array}{ll}
    0 & \text { if } x_{k-1}=0, \\
    \left\{ \frac{p_{k}}{x_{k-1}} \right\} & \text { if } x_{k-1} \neq 0,
    \end{array}
\right. \]

where \( \{x\} \) denotes the fractional part of \( x \). Find, with proof, all \( x_{0} \) satisfying \( 0 < x_{0} < 1 \) for which the sequence \( x_{0}, x_{1}, x_{2}, \ldots \) eventually becomes 0.

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