100 people are boarding an airplane (one at a time) that has 100 seats. Each person has an assigned seat.

The first person is confused, and randomly chooses a seat and sits down.

Each subsequent person will:

- Sit in their assigned seat if it is available.
- Randomly select and sit in an open seat if their assigned seat is taken.

What is the probability that the last person to board will sit in their assigned seat? When your answer is of the form \(\frac{p}{q}\), where \(p\) and \(q\) are positive, coprime integers, input \(p+q\).

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