Problem:
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

Solution in Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

N, A, B = gets.split(/ /).collect{|num| num.to_i}

M = 10 ** 9 + 7

prev_s = -10 ** 18 - 1
patterns_x = {prev_s => 1}
patterns_y = {}

N.times{
	s = gets.to_i
	new_patterns_x = {}
	new_patterns_y = {}
	patterns_x.each{|y_max, count|
		x_max = prev_s
		if x_ok = (s - x_max >= A)
			x_max = s - A
		end
		if y_ok = (s - y_max >= B)
			y_max = s - B
		end
		if x_ok
			if cur_count = new_patterns_x[y_max]
				new_patterns_x[y_max] = (cur_count + count) % M
			else
				new_patterns_x[y_max] = count
			end
		end
		if y_ok
			if cur_count = new_patterns_y[x_max]
				new_patterns_y[x_max] = (cur_count + count) % M
			else
				new_patterns_y[x_max] = count
			end
		end
	}
	patterns_y.each{|x_max, count|
		y_max = prev_s
		if x_ok = (s - x_max >= A)
			x_max = s - A
		end
		if y_ok = (s - y_max >= B)
			y_max = s - B
		end
		if x_ok
			if cur_count = new_patterns_x[y_max]
				new_patterns_x[y_max] = (cur_count + count) % M
			else
				new_patterns_x[y_max] = count
			end
		end
		if y_ok
			if cur_count = new_patterns_y[x_max]
				new_patterns_y[x_max] = (cur_count + count) % M
			else
				new_patterns_y[x_max] = count
			end
		end
	}
	patterns_x = new_patterns_x
	patterns_y = new_patterns_y
	prev_s = s
}
#p patterns
total_count = 0
patterns_x.each{|y_max, count|
	total_count = (total_count + count) % M
}
patterns_y.each{|x_max, count|
	total_count = (total_count + count) % M
}
puts total_count