
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
In a classroom, a student initially sits at the $(i, j)$ seat. If the new assigned seat is $(m, n)$, the student has moved by $[a, b] = [i - m, j - n]$, and $a + b$ is called the student's position number. The sum of all students' position numbers is denoted as $S$. Find the difference between the maximum and minimum possible values of $S$.

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