
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
At the boarding of an airplane, a queue of \( p \) passengers forms, each having a ticket for one of the \( n \) seats. The first in line is a crazy old woman. She rushes into the cabin and sits in a random seat (which might be her own). Then the remaining passengers take their seats one by one; if their seat is already occupied, they sit randomly in one of the remaining free seats. What is the probability that the last passenger will take their own seat?

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