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 Rust (1.15.1) (judge verdict: Accepted)

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String {
            bytes
                .by_ref()
                .map(|r| r.unwrap() as char)
                .skip_while(|c| c.is_whitespace())
                .take_while(|c| !c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, [ $t:tt ]) => {
        {
            let len = read_value!($next, usize);
            (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
        }
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_imports)]
use std::cmp::{min, max};

fn main() {
    input!{
        n: usize,
        a: usize,
        b: usize,
        v: [usize; n]
    }
    let (a, b) = (max(a, b), min(a, b));
    if n == 1 {
        println!("2");
        return;
    }
    for i in 0..n - 2 {
        if v[i + 2] - v[i] < b {
            println!("0");
            return;
        }
    }
    let d = 1000000007;
    let mut prev = vec![0; n];
    let mut p = 0;
    for i in 1..n {
        while v[p] + a <= v[i] {
            p += 1;
        }
        prev[i] = p;
    }
    let mut segt = SegT::new(n + 1, 0, |x, y| (x + y) % d, |x, y| (x + y) % d);
    segt.update(0, 1);
    segt.update(1, 1);
    let mut l = 0;
    for i in 1..n {
        let t = segt.get(l, prev[i]);
        segt.update(i + 1, t);
        if v[i] - v[i - 1] < b {
            l = i;
        }
    }
    println!("{}", segt.get(l, n));
}

struct SegT<T, F, G> where T: Copy, F: FnMut(T, T) -> T, G: FnMut(T, T) -> T {
    n: usize,
    v: Vec<T>,
    def: T,
    upf: F,
    getf: G
}

impl<T, F, G> SegT<T, F, G> where T: Copy, F: FnMut(T, T) -> T, G: FnMut(T, T) -> T {
    fn new(n: usize, def: T, upf: F, getf: G) -> SegT<T, F, G> {
        let mut s = 1;
        while s < n {
            s *= 2;
        }
        SegT{n: s, v: vec![def; 2 * s - 1], def: def, upf: upf, getf: getf}
    }

    fn update(&mut self, i: usize, x: T) {
        let n = self.n;
        self.update_rec(i, x, 0, 0, n - 1);
    }

    fn update_rec(&mut self, i: usize, x: T, p: usize, pl: usize, pr: usize) {
        if pl > i || pr < i {
            return;
        } else if p == i + self.n - 1 {
            self.v[p] = (self.upf)(self.v[p], x);
        } else {
            self.update_rec(i, x, 2 * p + 1, pl, (pl + pr - 1) / 2);
            self.update_rec(i, x, 2 * p + 2, (pl + pr + 1) / 2, pr);
            self.v[p] = (self.getf)(self.v[2 * p + 1], self.v[2 * p + 2]);
        }
    }

    fn get(&mut self, l: usize, r: usize) -> T {
        let n = self.n;
        self.get_rec(l, r, 0, 0, n - 1)
    }

    fn get_rec(&mut self, l: usize, r: usize, p: usize, pl: usize, pr: usize) -> T {
        if pl > r || pr < l {
            self.def
        } else if l <= pl && pr <= r {
            self.v[p]
        } else {
            let a = self.get_rec(l, r, 2 * p + 1, pl, (pl + pr - 1) / 2);
            let b = self.get_rec(l, r, 2 * p + 2, (pl + pr + 1) / 2, pr);
            (self.getf)(a, b)
        }
    }
}