You are given an array of integers $$$a_1, a_2, \ldots, a_n$$$.
You can do the following operation any number of times (possibly zero):
What is the minimum number of operations needed to turn $$$a$$$ into an arithmetic progression? The array $$$a$$$ is an arithmetic progression if $$$a_{i+1}-a_i=a_i-a_{i-1}$$$ for any $$$2 \leq i \leq n-1$$$.
The first line contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$).
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^5$$$).
Print a single integer: the minimum number of operations needed to turn $$$a$$$ into an arithmetic progression.
9 3 2 7 8 6 9 5 4 1
6
14 19 2 15 8 9 14 17 13 4 14 4 11 15 7
10
10 100000 1 60000 2 20000 4 8 16 32 64
7
4 10000 20000 10000 1
2
In the first test, you can get the array $$$a = [11, 10, 9, 8, 7, 6, 5, 4, 3]$$$ by performing $$$6$$$ operations:
$$$a$$$ is an arithmetic progression: in fact, $$$a_{i+1}-a_i=a_i-a_{i-1}=-1$$$ for any $$$2 \leq i \leq n-1$$$.
There is no sequence of less than $$$6$$$ operations that makes $$$a$$$ an arithmetic progression.
In the second test, you can get the array $$$a = [-1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38]$$$ by performing $$$10$$$ operations.
In the third test, you can get the array $$$a = [100000, 80000, 60000, 40000, 20000, 0, -20000, -40000, -60000, -80000]$$$ by performing $$$7$$$ operations.
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