Score : $400$ points

Problem Statement

You are given a sequence of $N$ positive integers: $A=(A_1,A_2, \dots A_N)$. You can do the operation below zero or more times. At least how many operations are needed to make $A$ a palindrome?

• Choose a pair $(x,y)$ of positive integers, and replace every occurrence of $x$ in $A$ with $y$.

Here, we say $A$ is a palindrome if and only if $A_i=A_{N+1-i}$ holds for every $i$ ($1 \le i \le N$).

Constraints

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

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer as an integer.

8
1 5 3 2 5 2 3 1

2

• Initially, we have $A=(1,5,3,2,5,2,3,1)$.
• After replacing every occurrence of $3$ in $A$ with $2$, we have $A=(1,5,2,2,5,2,2,1)$.
• After replacing every occurrence of $2$ in $A$ with $5$, we have $A=(1,5,5,5,5,5,5,1)$.

In this way, we can make $A$ a palindrome in two operations, which is the minimum needed.

7
1 2 3 4 1 2 3

1

1
200000

0

$A$ may already be a palindrome in the beginning.