
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 that go from the first floor to the last floor. On each staircase, between every two floors, there is a door at an intermediate landing that separates the floors (it is possible to access the floors from the staircase even if the door is locked). The superintendent decided that too many open doors is not good and therefore locked exactly half of the doors, choosing them randomly. What is the probability that it is possible to go from the first floor to the last floor using only open doors?

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