Squares
Description
There are $n$ squares drawn from left to right on the floor. The $i$-th square has three integers $p_i,a_i,b_i$, written on it. The sequence $p_1,p_2,\dots,p_n$ forms a permutation.
Each round you will start from the leftmost square $1$ and jump to the right. If you are now on the $i$-th square, you can do one of the following two operations:
- Jump to the $i+1$-th square and pay the cost $a_i$. If $i=n$, then you can end the round and pay the cost $a_i$.
- Jump to the $j$-th square and pay the cost $b_i$, where $j$ is the leftmost square that satisfies $j > i, p_j > p_i$. If there is no such $j$ then you can end the round and pay the cost $b_i$.
There are $q$ rounds in the game. To make the game more difficult, you need to maintain a square set $S$ (initially it is empty). You must pass through these squares during the round (other squares can also be passed through). The square set $S$ for the $i$-th round is obtained by adding or removing a square from the square set for the $(i-1)$-th round.
For each round find the minimum cost you should pay to end it.
The first line contains two integers $n$, $q$ ($1\le n,q\le 2 \cdot 10^5$) — the number of squares and the number of rounds.
The second line contains $n$ distinct integers $p_1,p_2,\dots,p_n$ ($1\le p_i\le n$). It is guaranteed that the sequence $p_1,p_2,\dots,p_n$ forms a permutation.
The third line contains $n$ integers $a_1,a_2,\ldots,a_n$ ($-10^9\le a_i\le 10^9$).
The fourth line contains $n$ integers $b_1,b_2,\dots,b_n$ ($-10^9\le b_i\le 10^9$).
Then $q$ lines follow, $i$-th of them contains a single integer $x_i$ ($1\le x_i\le n$). If $x_i$ was in the set $S$ on the $(i-1)$-th round you should remove it, otherwise, you should add it.
Print $q$ lines, each of them should contain a single integer — the minimum cost you should pay to end the corresponding round.
Input
The first line contains two integers $n$, $q$ ($1\le n,q\le 2 \cdot 10^5$) — the number of squares and the number of rounds.
The second line contains $n$ distinct integers $p_1,p_2,\dots,p_n$ ($1\le p_i\le n$). It is guaranteed that the sequence $p_1,p_2,\dots,p_n$ forms a permutation.
The third line contains $n$ integers $a_1,a_2,\ldots,a_n$ ($-10^9\le a_i\le 10^9$).
The fourth line contains $n$ integers $b_1,b_2,\dots,b_n$ ($-10^9\le b_i\le 10^9$).
Then $q$ lines follow, $i$-th of them contains a single integer $x_i$ ($1\le x_i\le n$). If $x_i$ was in the set $S$ on the $(i-1)$-th round you should remove it, otherwise, you should add it.
Output
Print $q$ lines, each of them should contain a single integer — the minimum cost you should pay to end the corresponding round.
Samples
3 2
2 1 3
10 -5 4
3 -2 3
1
2
6
8
5 4
2 1 5 3 4
6 -5 3 -10 -1
0 3 2 7 2
1
2
3
2
-8
-7
-7
-8
Note
Let's consider the character $T$ as the end of a round. Then we can draw two graphs for the first and the second test.
In the first round of the first test, the set that you must pass through is $\{1\}$. The path you can use is $1\to 3\to T$ and its cost is $6$.
In the second round of the first test, the set that you must pass through is $\{1,2\}$. The path you can use is $1\to 2\to 3\to T$ and its cost is $8$.