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Contest: Task: Related: TaskB TaskD

Score : $1100$ points

Problem Statement

There is a set consisting of $N$ distinct integers. The $i$-th smallest element in this set is $S_i$. We want to divide this set into two sets, $X$ and $Y$, such that:

  • The absolute difference of any two distinct elements in $X$ is $A$ or greater.
  • The absolute difference of any two distinct elements in $Y$ is $B$ or greater.

How many ways are there to perform such division, modulo $10^9 + 7$? Note that one of $X$ and $Y$ may be empty.

Constraints

  • All input values are integers.
  • $1 ≦ N ≦ 10^5$
  • $1 ≦ A , B ≦ 10^{18}$
  • $0 ≦ S_i ≦ 10^{18}(1 ≦ i ≦ N)$
  • $S_i < S_{i+1}(1 ≦ i ≦ N - 1)$

Input

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

$N$ $A$ $B$
$S_1$
:
$S_N$

Output

Print the number of the different divisions under the conditions, modulo $10^9 + 7$.

Sample Input 1

5 3 7
1
3
6
9
12

Sample Output 1

5

There are five ways to perform division:

  • $X=${$1,6,9,12$}, $Y=${$3$}
  • $X=${$1,6,9$}, $Y=${$3,12$}
  • $X=${$3,6,9,12$}, $Y=${$1$}
  • $X=${$3,6,9$}, $Y=${$1,12$}
  • $X=${$3,6,12$}, $Y=${$1,9$}

Sample Input 2

7 5 3
0
2
4
7
8
11
15

Sample Output 2

4

Sample Input 3

8 2 9
3
4
5
13
15
22
26
32

Sample Output 3

13

Sample Input 4

3 3 4
5
6
7

Sample Output 4

0