The four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it vanishes, and a number of copies equal to the number on the face it landed on appear. For example:
- If it lands on face $0,$ it disappears.
- If it lands on face $1,$ nothing happens.
- If it lands on face $2$ or $3,$ there will be $2$ or $3$ copies of the die, respectively (including the original).

Suppose the die and all its copies are continually rolled. Let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

Let's think step by step and output the final answer within \boxed{}.
user