
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
The four faces of a tetrahedral die are labelled  $0, 1, 2,$  and  $3,$  and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled  $0,$  it disappears. If it lands on the face labelled  $1,$  nothing happens. If it lands on the face labelled  $2$  or  $3,$  there will then be  $2$  or  $3$  copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let  $p$  be the probability that they will all eventually disappear. Find  $\left\lfloor \frac{10}{p} \right\rfloor$ .

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