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G - Sort from 1 to 4

Score : $625$ points

Problem Statement

You are given a sequence $A=(A_1,A_2,\ldots,A_N)$ of length $N$, consisting of integers between $1$ and $4$.

Takahashi may perform the following operation any number of times (possibly zero):

• choose a pair of integer $(i,j)$ such that $1\leq i<j\leq N$, and swap $A_i$ and $A_j$.

Find the minimum number of operations required to make $A$ non-decreasing.
A sequence is said to be non-decreasing if and only if $A_i\leq A_{i+1}$ for all $1\leq i\leq N-1$.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $1\leq A_i \leq 4$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the minimum number of operations required to make $A$ non-decreasing in a single line.

6
3 4 1 1 2 4

3

One can make $A$ non-decreasing with the following three operations:

• Choose $(i,j)=(2,3)$ to swap $A_2$ and $A_3$, making $A=(3,1,4,1,2,4)$.
• Choose $(i,j)=(1,4)$ to swap $A_1$ and $A_4$, making $A=(1,1,4,3,2,4)$.
• Choose $(i,j)=(3,5)$ to swap $A_3$ and $A_5$, making $A=(1,1,2,3,4,4)$.

This is the minimum number of operations because it is impossible to make $A$ non-decreasing with two or fewer operations.
Thus, $3$ should be printed.

4
2 3 4 1

3