
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
Numbers \(1, 2, \ldots, 2010\) are placed on the circumference of a circle in some order. The numbers \(i\) and \(j\), where \(i \neq j\) and \(i, j \in \{1, 2, \ldots, 2010\}\), form a friendly pair if:
(i) \(i\) and \(j\) are not neighbors to each other, and
(ii) on one or both of the arcs connecting \(i\) and \(j\) along the circle, all numbers in between them are greater than both \(i\) and \(j\).

Determine the minimal number of friendly pairs.

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