
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
A building has $n$ floors and two staircases running from the first to the last floor. On each staircase, there is a door on the intermediate landing between each pair of floors; one can get from the staircase to a floor even if the door is locked. The custodian decided that having too many open doors was undesirable, so they locked exactly half of the doors, choosing them at random. What is the probability that it is possible to go from the first floor to the last floor, passing through only open doors?

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