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Ex - Monster

Score : $600$ points

Problem Statement

There are $N$ monsters along a number line. At the coordinate $i$ $(1\leq i\leq N)$ is a monster with a stamina of $A_i$.
Additionally, at the coordinate $i$, there is a permanent shield of a strength $B_i$.
This shield persists even when the monster at the same coordinate has a health of $0$ or below.

Takahashi, a magician, can perform the following operation any number of times.

• Choose integers $l$ and $r$ satisfying $1\leq l\leq r\leq N$.
• Then, consume $\max(B_l, B_{l+1}, \ldots, B_r)$ MP (magic points) to decrease by $1$ the stamina of each of the monsters at the coordinates $l,l+1,\ldots,r$.

When choosing $l$ and $r$, it is fine if some of the monsters at the coordinates $l,l+1,\ldots,r$ already have a stamina of $0$ or below.
Note, however, that the shields at all those coordinates still exist.

Takahashi wants to make the stamina of every monster $0$ or below.
Find the minimum total MP needed to achieve his objective.

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq A_i,B_i \leq 10^9$
• All values in the input are integers.

Input

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

$N$

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

$B_1$ $B_2$ $\ldots$ $B_N$

Output

Print the minimum total MP needed to achieve his objective.

5
4 3 5 1 2
10 40 20 60 50

210

Takahashi can achieve his objective as follows.

• Choose $(l,r)=(1,5)$. Consume $\max(10,40,20,60,50)=60$ MP, and the staminas of the monsters are $(3,2,4,0,1)$.
• Choose $(l,r)=(1,5)$. Consume $\max(10,40,20,60,50)=60$ MP, and the staminas of the monsters are $(2,1,3,-1,0)$.
• Choose $(l,r)=(1,3)$. Consume $\max(10,40,20)=40$ MP, and the staminas of the monsters are $(1,0,2,-1,0)$.
• Choose $(l,r)=(1,1)$. Consume $\max(10)=10$ MP, and the staminas of the monsters are $(0,0,2,-1,0)$.
• Choose $(l,r)=(3,3)$. Consume $\max(20)=20$ MP, and the staminas of the monsters are $(0,0,1,-1,0)$.
• Choose $(l,r)=(3,3)$. Consume $\max(20)=20$ MP, and the staminas of the monsters are $(0,0,0,-1,0)$.

Here, he consumes a total of $60+60+40+10+20+20=210$ MP, which is the minimum possible.

1
1000000000
1000000000

1000000000000000000

10
522 4575 6426 9445 8772 81 3447 629 3497 7202
7775 4325 3982 4784 8417 2156 1932 5902 5728 8537

77917796