
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
Fix an odd integer  $n > 1$ . For a permutation  $p$  of the set  $\{1,2,...,n\}$ , let S be the number of pairs of indices  $(i, j)$ ,  $1 \le i \le  j \le  n$ , for which  $p_i +p_{i+1} +...+p_j$  is divisible by  $n$ . Determine the maximum possible value of  $S$ .

Croatia

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