
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
Consider \( n \) disks \( C_{1}, C_{2}, \ldots, C_{n} \) in a plane such that for each \( 1 \leq i < n \), the center of \( C_{i} \) is on the circumference of \( C_{i+1} \), and the center of \( C_{n} \) is on the circumference of \( C_{1} \). Define the score of such an arrangement of \( n \) disks to be the number of pairs \((i, j)\) for which \( C_{i} \) properly contains \( C_{j} \). Determine the maximum possible score.

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