
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 \( n \geq 5 \) be an integer. There are \( n \) points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the \( i \)-th day, for \( 1 \leq i \leq n-3 \), before erasing that day's point, Tom writes down the positive integer \( v(i) \) such that the convex hull of the points at that moment has \( v(i) \) vertices. Finally, he writes down \( v(n-2) = 3 \). Find the greatest possible value that the expression 
 
\[ 
|v(1) - v(2)| + |v(2) - v(3)| + \ldots + |v(n-3) - v(n-2)| 
\]
 
can obtain among all possible initial configurations of \( n \) points and all possible Tom's moves.

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