
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 \( p_{1}, p_{2}, p_{3}, \cdots \) be the sequence of prime numbers arranged in increasing order. Let \( x_{0} \) be a real number between 0 and 1. For a positive integer \( k \), define
\[ 
x_{k} = \begin{cases} 
0 & \text{if } x_{k-1}=0 \\
\left\{\frac{p_{k}}{x_{k-1}}\right\} & \text{if } x_{k-1} \neq 0 
\end{cases} 
\]
where \(\{x\}\) denotes the fractional part of \( x \). Find all values of \( x_{0} \) satisfying \( 0 < x_{0} < 1 \) such that the sequence \( x_{0}, x_{1}, x_{2}, \cdots \) eventually contains 0, and provide a proof for your solution.

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