
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
In a building with \( n \) floors and two staircases going from the first to the last floor, there is a door on each intermediate landing between every two floors. The doors separate the floors (you can access the floor from the staircase, even if the door is locked). The building superintendent decided that too many open doors are undesirable and locked exactly half of the doors, choosing the doors randomly. What is the probability that it is possible to ascend from the first floor to the last floor, passing only through open doors?

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