F2. Wine Factory (Hard Version)
time limit per test
5 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

This is the hard version of the problem. The only difference between the two versions is the constraint on $c_i$ and $z$. You can make hacks only if both versions of the problem are solved.

There are three arrays $a$, $b$ and $c$. $a$ and $b$ have length $n$ and $c$ has length $n-1$. Let $W(a,b,c)$ denote the liters of wine created from the following process.

Create $n$ water towers. The $i$-th water tower initially has $a_i$ liters of water and has a wizard with power $b_i$ in front of it. Furthermore, for each $1 \le i \le n - 1$, there is a valve connecting water tower $i$ to $i + 1$ with capacity $c_i$.

For each $i$ from $1$ to $n$ in this order, the following happens:

1. The wizard in front of water tower $i$ removes at most $b_i$ liters of water from the tower and turns the removed water into wine.
2. If $i \neq n$, at most $c_i$ liters of the remaining water left in water tower $i$ flows through the valve into water tower $i + 1$.

There are $q$ updates. In each update, you will be given integers $p$, $x$, $y$ and $z$ and you will update $a_p := x$, $b_p := y$ and $c_p := z$. After each update, find the value of $W(a,b,c)$. Note that previous updates to arrays $a$, $b$ and $c$ persist throughout future updates.

Input

The first line contains two integers $n$ and $q$ ($2 \le n \le 5\cdot 10^5$, $1 \le q \le 5\cdot 10^5$) — the number of water towers and the number of updates.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the number of liters of water in water tower $i$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($0 \le b_i \le 10^9$) — the power of the wizard in front of water tower $i$.

The fourth line contains $n - 1$ integers $c_1, c_2, \ldots, c_{n - 1}$ ($0 \le c_i \color{red}{\le} 10^{18}$) — the capacity of the pipe connecting water tower $i$ to $i + 1$.

Each of the next $q$ lines contains four integers $p$, $x$, $y$ and $z$ ($1 \le p \le n$, $0 \le x, y \le 10^9$, $0 \le z \color{red}{\le} 10^{18}$) — the updates done to arrays $a$, $b$ and $c$.

Note that $c_n$ does not exist, so the value of $z$ does not matter when $p = n$.

Output

Print $q$ lines, each line containing a single integer representing $W(a, b, c)$ after each update.

Examples
Input
4 33 3 3 31 4 2 85 2 14 3 8 10000000002 5 1 13 0 0 0
Output
11
8
5

Input
5 510 3 8 9 23 4 10 8 16 5 9 25 4 9 11 1 1 12 7 4 84 1 1 11 8 3 3
Output
31
25
29
21
23

Note

The first update does not make any modifications to the arrays.

• When $i = 1$, there are $3$ liters of water in tower 1 and $1$ liter of water is turned into wine. The remaining $2$ liters of water flow into tower 2.
• When $i = 2$, there are $5$ liters of water in tower 2 and $4$ liters of water is turned into wine. The remaining $1$ liter of water flows into tower 3.
• When $i = 3$, there are $4$ liters of water in tower 3 and $2$ liters of water is turned into wine. Even though there are $2$ liters of water remaining, only $1$ liter of water can flow into tower 4.
• When $i = 4$, there are $4$ liters of water in tower 4. All $4$ liters of water are turned into wine.

Hence, $W(a,b,c)=1 + 4 + 2 + 4 = 11$ after the first update.

The second update modifies the arrays to $a = [3, 5, 3, 3]$, $b = [1, 1, 2, 8]$, and $c = [5, 1, 1]$.

• When $i = 1$, there are $3$ liters of water in tower 1 and $1$ liter of water is turned into wine. The remaining $2$ liters of water flow into tower 2.
• When $i = 2$, there are $7$ liters of water in tower 2 and $1$ liter of water is turned into wine. Even though there are $6$ liters of water remaining, only $1$ liter of water can flow to tower 3.
• When $i = 3$, there are $4$ liters of water in tower 3 and $2$ liters of water is turned into wine. Even though there are $2$ liters of water remaining, only $1$ liter of water can flow into tower 4.
• When $i = 4$, there are $4$ liters of water in tower 4. All $4$ liters of water are turned into wine.

Hence, $W(a,b,c)=1 + 1 + 2 + 4 = 8$ after the second update.

The third update modifies the arrays to $a = [3, 5, 0, 3]$, $b = [1, 1, 0, 8]$, and $c = [5, 1, 0]$.

• When $i = 1$, there are $3$ liters of water in tower 1 and $1$ liter of water is turned into wine. The remaining $2$ liters of water flow into tower 2.
• When $i = 2$, there are $7$ liters of water in tower 2 and $1$ liter of water is turned into wine. Even though there are $6$ liters of water remaining, only $1$ liter of water can flow to tower 3.
• When $i = 3$, there is $1$ liter of water in tower 3 and $0$ liters of water is turned into wine. Even though there is $1$ liter of water remaining, no water can flow to tower 4.
• When $i = 4$, there are $3$ liters of water in tower 4. All $3$ liters of water are turned into wine.

Hence, $W(a,b,c)=1 + 1 + 0 + 3 = 5$ after the third update.