
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
There is a public phone booth with a telephone inside. When observing the flow of people using this phone, assume that the probability of $n$ people using or waiting to use the phone at a certain moment is $P(n)$, and $P(n)$ is independent of the time $t$. It is statistically found that $P(n) = \begin{cases} (\frac{1}{2})^{n}\cdot P(0), & 1\leq n \leq 6 \\ 0, & n\geq 7 \end{cases}$. Then, at a certain moment, the probability $P(0)$ that there is no one in this public phone booth is \_\_\_\_\_\_.

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