
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  $C_{1}; C_{2}; ... ; C_{n}$  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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