
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
Another participant in the contest of meaningless activities marked the centers of 13 cells in a grid rectangle of size \((N-1) \times (N+1)\) such that the distance between any two marked points is greater than 2. What is the smallest possible value of \(N\)?

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