
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
An accident occurred in the reactor of a research spaceship, causing toxic substances to leak out. All corridors between compartments are equipped with airtight doors, but there is no time to close individual doors. However, the captain can issue the command "Close $N$ doors", after which the ship's artificial intelligence will randomly close $N$ doors. What is the smallest value of $N$ that guarantees at least one of the ship's compartments remains habitable?

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