
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 telephone booth with a telephone inside. While observing the flow of people using the telephone, let $P(n)$ denote the probability that there are $n$ people using or waiting to use the telephone at a certain moment, and $P(n)$ is independent of the time $t$. It is found that $P(n)= \begin{cases} (\frac{1}{2})^{n} \cdot P(0), 1 \leqslant n \leqslant 6 \\ 0, n \geqslant 7 \end{cases}$. Determine the value of $P(0)$, the probability that there is no one in the telephone booth at a certain moment.

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