
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
NASA has proposed populating Mars with 2,004 settlements. The only way to get from one settlement to another will be by a connecting tunnel. A bored bureaucrat draws on a map of Mars, randomly placing \( N \) tunnels connecting the settlements in such a way that no two settlements have more than one tunnel connecting them. What is the smallest value of \( N \) that guarantees that, no matter how the tunnels are drawn, it will be possible to travel between any two settlements?

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