
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
Let  $n \geq 3$  be an integer. John and Mary play the following game: First John labels the sides of a regular  $n$ -gon with the numbers  $1, 2,\ldots, n$  in whatever order he wants, using each number exactly once. Then Mary divides this  $n$ -gon into triangles by drawing  $n-3$  diagonals which do not intersect each other inside the  $n$ -gon. All these diagonals are labeled with number  $1$ . Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those  $n - 2$  products.

Determine the value of  $S$  if Mary wants the number  $S$  to be as small as possible and John wants  $S$  to be as large as possible and if they both make the best possible choices.

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