
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  $n>5$  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  $1<i<n-3$ , 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.

*Remark*. A convex hull of a finite set of points in the plane is the smallest convex polygon containing all the points of the set (inside it or on the boundary).

*Ivan Novak, Namik Agić*

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