D. Almost Arithmetic Progression

Polycarp likes arithmetic progressions. A sequence $[ a_1, a_2, dots.h, a_n ]$ is called an arithmetic progression if for each $i$ ($1 lt.eq i < n$) the value $a_(i + 1) -a_i$ is the same. For example, the sequences $[ 42 ]$, $[ 5, 5, 5 ]$, $[ 2, 11, 20, 29 ]$ and $[ 3, 2, 1, 0 ]$ are arithmetic progressions, but $[ 1, 0, 1 ]$, $[ 1, 3, 9 ]$ and $[ 2, 3, 1 ]$ are not. It follows from the definition that any sequence of length one or two is an arithmetic progression. Polycarp found some sequence of positive integers $[ b_1, b_2, dots.h, b_n ]$. He agrees to change each element by at most one. In the other words, for each element there are exactly three options: an element can be decreased by $1$, an element can be increased by $1$, an element can be left unchanged. Determine a minimum possible number of elements in $b$ which can be changed (by exactly one), so that the sequence $b$ becomes an arithmetic progression, or report that it is impossible. It is possible that the resulting sequence contains element equals $0$. The first line contains a single integer $n$ $(1 lt.eq n lt.eq 100 thin 000)$ — the number of elements in $b$. The second line contains a sequence $b_1, b_2, dots.h, b_n$ $(1 lt.eq b_i lt.eq 10^9)$. If it is impossible to make an arithmetic progression with described operations, print _-1_. In the other case, print non-negative integer — the minimum number of elements to change to make the given sequence becomes an arithmetic progression. The only allowed operation is to add/to subtract one from an element (can't use operation twice to the same position). ## Input The first line contains a single integer $n$ $(1 lt.eq n lt.eq 100 thin 000)$ — the number of elements in $b$.The second line contains a sequence $b_1, b_2, dots.h, b_n$ $(1 lt.eq b_i lt.eq 10^9)$. ## Output If it is impossible to make an arithmetic progression with described operations, print _-1_. In the other case, print non-negative integer — the minimum number of elements to change to make the given sequence becomes an arithmetic progression. The only allowed operation is to add/to subtract one from an element (can't use operation twice to the same position). [samples] ## Note In the first example Polycarp should increase the first number on $1$, decrease the second number on $1$, increase the third number on $1$, and the fourth number should left unchanged. So, after Polycarp changed three elements by one, his sequence became equals to $[ 25, 20, 15, 10 ]$, which is an arithmetic progression.In the second example Polycarp should not change anything, because his sequence is an arithmetic progression.In the third example it is impossible to make an arithmetic progression.In the fourth example Polycarp should change only the first element, he should decrease it on one. After that his sequence will looks like $[ 0, 3, 6, 9, 12 ]$, which is an arithmetic progression.

**Definitions** Let $ n \in \mathbb{Z} $, $ n \geq 1 $, be the length of the sequence. Let $ B = (b_1, b_2, \dots, b_n) $ be a sequence of positive integers, where $ b_i \in \mathbb{Z} $ and $ 1 \leq b_i \leq 10^9 $ for all $ i \in \{1, \dots, n\} $. An arithmetic progression (AP) is a sequence $ A = (a_1, a_2, \dots, a_n) $ such that $ a_{i+1} - a_i = d $ for all $ i \in \{1, \dots, n-1\} $, for some common difference $ d \in \mathbb{R} $. Each element $ b_i $ may be transformed to one of three values: $ b_i - 1 $, $ b_i $, or $ b_i + 1 $. **Constraints** 1. $ 1 \leq n \leq 100{,}000 $ 2. $ 1 \leq b_i \leq 10^9 $ for all $ i \in \{1, \dots, n\} $ 3. The resulting sequence $ A = (a_1, \dots, a_n) $ must satisfy $ a_{i+1} - a_i = d $ for all $ i \in \{1, \dots, n-1\} $, for some fixed $ d \in \mathbb{R} $. 4. For each $ i $, $ a_i \in \{b_i - 1, b_i, b_i + 1\} $. **Objective** Find the minimum number of indices $ i \in \{1, \dots, n\} $ such that $ a_i \neq b_i $, over all possible arithmetic progressions $ A $ achievable under the constraints. If no such arithmetic progression exists, return $ -1 $. Equivalently: Minimize $ \sum_{i=1}^n \mathbf{1}_{a_i \neq b_i} $ subject to: - $ a_i \in \{b_i - 1, b_i, b_i + 1\} $ for all $ i $, - $ \exists\, d \in \mathbb{R} $ such that $ a_{i+1} - a_i = d $ for all $ i \in \{1, \dots, n-1\} $.