
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]


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For each positive integer \( n \), consider a cinema with \( n \) seats in a row, numbered left to right from 1 up to \( n \). There is a cup holder between any two adjacent seats, and there is a cup holder on the right of seat \( n \). So seat 1 is next to one cup holder, while every other seat is next to two cup holders. There are \( n \) people, each holding a drink, waiting in a line to sit down. In turn, each person chooses an available seat uniformly at random and carries out the following:
- If they sit next to two empty cup holders, then they place their drink in the left cup holder with probability \(\frac{1}{2}\) or in the right cup holder with probability \(\frac{1}{2}\).
- If they sit next to one empty cup holder, then they place their drink in that empty cup holder.
- If they sit next to zero empty cup holders, then they hold their drink in their hands.

Let \( p_{n} \) be the probability that all \( n \) people place their drink in a cup holder. Determine \( p_{1} + p_{2} + p_{3} + \cdots \).

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