201313: [AtCoder]ARC131 D - AtArcher

Problem Statement

Ringo is participating in an archery contest AtArcher.

In AtArcher, a participant shoots $N$ arrows at a target on a number line to compete for the total score. The center of the target is at coordinate $0$. Based on where the arrow hits, the score of the shot is defined as follows.

• For $i = 0, 1, \dots, M-1$, if the arrow hits a point whose distance from the center of the target is between $r_i$ and $r_{i+1}$, the score is $s_i$. If the distance is greater than $r_M$, the score is $0$. If the arrow hits the boundary, the higher score is applied.
• The closer to the center, the higher the score is. In other words, the following is satisfied.
  • $0 = r_0 \lt r_1 \lt \cdots \lt r_{M-1} \lt r_M$
  • $s_0 \gt s_1 \gt \cdots \gt s_{M-1} \gt 0$

For example, the figure below shows the score for a shot when $r = (0, 2, 7, 9), s = (100, 70, 30)$.

Additionally, AtArcher has one special rule: the distance between any two arrows must always be at least $D$. Violating this rule disqualifies the participant, making the total score $0$.

What is the maximum total score that Ringo can get from all the shots?

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq M \leq 10^5$
• $1 \leq D \leq 10^6$
• $0 = r_0 \lt r_1 \lt \cdots \lt r_{M-1} \lt r_M \leq 10^{11}$
• $10^{11} \geq s_0 \gt s_1 \gt \cdots \gt s_{M-1} \gt 0$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $D$
$r_0$ $r_1$ $\cdots$ $r_{M-1}$ $r_M$
$s_0$ $s_1$ $\cdots$ $s_{M-1}$

Output

Print the answer.

Sample Input 1

3 3 3
0 2 7 9
100 70 30

Sample Output 1

270

This sample input corresponds to the case in the Problem Statement, with $D = 3$.

For example, if the $N = 3$ arrows hit the coordinates $-6, -2, 1$, they score $70, 100, 100$, respectively, for a total score of $270$, which is the maximum achievable.

Note that you cannot hit the $100$-point area with all the arrows to score $300$, because the distance between any two arrows must be at least $D = 3$, or you will be disqualified and score $0$.

Sample Input 2

3 3 8
0 2 7 9
100 70 30

Sample Output 2

200

This sample input corresponds to the case in the Problem Statement, with $D = 8$.

For example, if the $N = 3$ arrows hit the coordinates $-7, 1, 9$, they score $70, 100, 30$, respectively, for a total score of $200$, which is the maximum achievable.

Sample Input 3

7 5 47
0 10 40 100 160 220
50 25 9 6 3

Sample Output 3

111

For example, if you shoot the arrows as shown in the following figure, you will score $111$ in total, which is the maximum.

Sample Input 4

100 1 5
0 7
100000000000

Sample Output 4

300000000000

You can shoot $N = 100$ arrows, but in order to avoid disqualification, at most $3$ arrows can hit the area with a positive score.

Sample Input 5

15 10 85
0 122 244 366 488 610 732 854 976 1098 1220
10 9 8 7 6 5 4 3 2 1

Sample Output 5

119