
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
There are  $32$  students in the class with  $10$  interesting group. Each group contains exactly  $16$  students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their  $interests-disparity$ . Define  $S$  as the sum of the  $interests-disparity$  of all the couples,  $\binom{32}{2}\left ( =\: 496  \right )$  ones in total. Determine the minimal possible value of  $S$ .

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