Problem:
Score : 400 points

Problem Statement

A maze is composed of a grid of H \times W squares - H vertical, W horizontal.

The square at the i-th row from the top and the j-th column from the left - (i,j) - is a wall if S_{ij} is # and a road if S_{ij} is ..

There is a magician in (C_h,C_w). He can do the following two kinds of moves:

Move A: Walk to a road square that is vertically or horizontally adjacent to the square he is currently in.

Move B: Use magic to warp himself to a road square in the 5\times 5 area centered at the square he is currently in.

In either case, he cannot go out of the maze.

At least how many times does he need to use the magic to reach (D_h, D_w)?

Constraints

1 \leq H,W \leq 10^3

1 \leq C_h,D_h \leq H

1 \leq C_w,D_w \leq W

S_{ij} is # or ..

S_{C_h C_w} and S_{D_h D_w} are ..

(C_h,C_w) \neq (D_h,D_w)

Input

Input is given from Standard Input in the following format:

H W
C_h C_w
D_h D_w
S_{11}\ldots S_{1W}
\vdots
S_{H1}\ldots S_{HW}

Output

Print the minimum number of times the magician needs to use the magic. If he cannot reach (D_h,D_w), print -1 instead.

Sample Input 1

4 4
1 1
4 4
..#.
..#.
.#..
.#..

Sample Output 1

1

For example, by walking to (2,2) and then using the magic to travel to (4,4), just one use of magic is enough.

Note that he cannot walk diagonally.

Sample Input 2

4 4
1 4
4 1
.##.
####
####
.##.

Sample Output 2

-1

He cannot move from there.

Sample Input 3

4 4
2 2
3 3
....
....
....
....

Sample Output 3

0

No use of magic is needed.

Sample Input 4

4 5
1 2
2 5
#.###
####.
#..##
#..##

Sample Output 4

2

Solution in Rust (1.42.0) (judge verdict: Wrong Answer)

#[allow(dead_code)]
fn main() {
    let stdin = stdin();
    solve(StdinReader::new(stdin.lock()));
}

pub fn solve<R: BufRead>(mut reader: StdinReader<R>) {
    let (h, w) = reader.u2();
    let (ch, cw) = reader.u2();
    let (dh, dw) = reader.u2();
    let s = reader.cmap(h);

    let g = Grid::new(h, w, s);
    let mut d = vec![h * w; h * w];

    let mut q = VecDeque::new();
    q.push_back(g.key(cw - 1, ch - 1));
    d[g.key(cw - 1, ch - 1)] = 0;
    let mut s = HashSet::new();
    s.insert(g.key(cw - 1, ch - 1));
    loop {
        while q.len() > 0 {
            let from = q.pop_front().unwrap();
            let nei = g.neighbor(from);
            for to in nei {
                if d[to] < d[from] {
                    continue;
                }
                if g.get(to) == g.get(from) {
                    if d[from] < d[to] {
                        d[to] = d[from];
                        q.push_back(to);
                        s.insert(to);
                    }
                }
            }
        }
        if s.is_empty() {
            break;
        }
        for &from in &s {
            let warp = g.warp(from);
            for to in warp {
                if d[to] <= d[from] {
                    continue;
                }
                if g.get(to) == g.get(from) {
                    if d[from] + 1 < d[to] {
                        d[to] = d[from] + 1;
                        q.push_back(to);
                    }
                }
            }
        }
        s = HashSet::new();
    }
    if d[g.key(dw - 1, dh - 1)] == h * w {
        println!("{}", -1);
    } else {
        println!("{}", d[g.key(dw - 1, dh - 1)]);
    }
}

#[allow(unused_imports)]
use grid::*;

#[allow(dead_code)]
mod grid {
    #[derive(Debug)]
    pub struct Grid<T> {
        pub h: usize,
        pub w: usize,
        pub max: usize,
        pub map: Vec<T>,
    }

    impl<T: Clone> Grid<T> {
        pub fn new(h: usize, w: usize, map: Vec<Vec<T>>) -> Grid<T> {
            let mut flat = Vec::new();
            for r in map {
                for c in r {
                    flat.push(c);
                }
            }
            Grid {
                h: h,
                w: w,
                max: h * w,
                map: flat,
            }
        }
        pub fn key(&self, x: usize, y: usize) -> usize {
            y * self.w + x
        }
        pub fn xy(&self, k: usize) -> (usize, usize) {
            (self.x(k), self.y(k))
        }
        pub fn x(&self, k: usize) -> usize {
            k % self.w
        }
        pub fn y(&self, k: usize) -> usize {
            k / self.w
        }
        pub fn get(&self, key: usize) -> &T {
            &self.map[key]
        }
        pub fn set(&mut self, key: usize, value: T) {
            self.map[key] = value;
        }
        pub fn neighbor(&self, key: usize) -> Vec<usize> {
            let mut ret = self.one_way(key);
            if self.x(key) > 0 {
                ret.push(key - 1);
            }
            if self.y(key) > 0 {
                ret.push(key - self.w);
            }
            ret
        }
        pub fn one_way(&self, key: usize) -> Vec<usize> {
            let mut ret = Vec::new();
            if self.x(key) + 1 < self.w {
                ret.push(key + 1);
            }
            if self.y(key) + 1 < self.h {
                ret.push(key + self.w);
            }
            ret
        }

        pub fn warp(&self, key: usize) -> Vec<usize> {
            let mut ret = Vec::new();
            for i in 0..5 {
                for j in 0..5 {
                    if (i == 2 && j == 2)
                        || self.x(key) + i <= 1
                        || self.y(key) + j <= 1
                        || self.x(key) + i >= self.w + 2
                        || self.y(key) + j >= self.h + 2
                    {
                        continue;
                    }
                    ret.push(key + i - 2 + self.w * j - 2 * self.w)
                }
            }
            ret
        }
    }
}

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use std::{cmp::*, collections::*, io::*, num::*, str::*};
#[allow(unused_imports)]
use stdin_reader::StdinReader;

#[allow(dead_code)]
pub mod stdin_reader {
    use std::{fmt::Debug, io::*, str::*};

    pub struct StdinReader<R: BufRead> {
        reader: R,
        buf: Vec<u8>,
        // Should never be empty
        pos: usize, // Should never be out of bounds as long as the input ends with '\n'
    }

    impl<R: BufRead> StdinReader<R> {
        pub fn new(reader: R) -> StdinReader<R> {
            let (buf, pos) = (Vec::new(), 0);
            StdinReader { reader, buf, pos }
        }

        pub fn n<T: FromStr>(&mut self) -> T
        where
            T::Err: Debug,
        {
            if self.buf.is_empty() {
                self._read_next_line();
            }
            let mut start = None;
            while self.pos != self.buf.len() {
                match (self.buf[self.pos], start.is_some()) {
                    (b' ', true) | (b'\n', true) => break,
                    (_, true) | (b' ', false) => self.pos += 1,
                    (b'\n', false) => self._read_next_line(),
                    (_, false) => start = Some(self.pos),
                }
            }
            match start {
                Some(s) => from_utf8(&self.buf[s..self.pos]).unwrap().parse().unwrap(),
                None => panic!("入力された数を超えた読み込みが発生しています"),
            }
        }

        fn _read_next_line(&mut self) {
            self.pos = 0;
            self.buf.clear();
            if self.reader.read_until(b'\n', &mut self.buf).unwrap() == 0 {
                panic!("Reached EOF");
            }
        }

        pub fn str(&mut self) -> String {
            self.n()
        }
        pub fn s(&mut self) -> Vec<char> {
            self.n::<String>().chars().collect()
        }
        pub fn i(&mut self) -> i64 {
            self.n()
        }
        pub fn i2(&mut self) -> (i64, i64) {
            (self.n(), self.n())
        }
        pub fn i3(&mut self) -> (i64, i64, i64) {
            (self.n(), self.n(), self.n())
        }
        pub fn u(&mut self) -> usize {
            self.n()
        }
        pub fn u2(&mut self) -> (usize, usize) {
            (self.n(), self.n())
        }
        pub fn u3(&mut self) -> (usize, usize, usize) {
            (self.n(), self.n(), self.n())
        }
        pub fn u4(&mut self) -> (usize, usize, usize, usize) {
            (self.n(), self.n(), self.n(), self.n())
        }
        pub fn u5(&mut self) -> (usize, usize, usize, usize, usize) {
            (self.n(), self.n(), self.n(), self.n(), self.n())
        }
        pub fn u6(&mut self) -> (usize, usize, usize, usize, usize, usize) {
            (self.n(), self.n(), self.n(), self.n(), self.n(), self.n())
        }
        pub fn f(&mut self) -> f64 {
            self.n()
        }
        pub fn f2(&mut self) -> (f64, f64) {
            (self.n(), self.n())
        }
        pub fn c(&mut self) -> char {
            self.n::<String>().pop().unwrap()
        }
        pub fn iv(&mut self, n: usize) -> Vec<i64> {
            (0..n).map(|_| self.i()).collect()
        }
        pub fn iv2(&mut self, n: usize) -> Vec<(i64, i64)> {
            (0..n).map(|_| self.i2()).collect()
        }
        pub fn iv3(&mut self, n: usize) -> Vec<(i64, i64, i64)> {
            (0..n).map(|_| self.i3()).collect()
        }
        pub fn uv(&mut self, n: usize) -> Vec<usize> {
            (0..n).map(|_| self.u()).collect()
        }
        pub fn uv2(&mut self, n: usize) -> Vec<(usize, usize)> {
            (0..n).map(|_| self.u2()).collect()
        }
        pub fn uv3(&mut self, n: usize) -> Vec<(usize, usize, usize)> {
            (0..n).map(|_| self.u3()).collect()
        }
        pub fn uv4(&mut self, n: usize) -> Vec<(usize, usize, usize, usize)> {
            (0..n).map(|_| self.u4()).collect()
        }
        pub fn fv(&mut self, n: usize) -> Vec<f64> {
            (0..n).map(|_| self.f()).collect()
        }
        pub fn cmap(&mut self, h: usize) -> Vec<Vec<char>> {
            (0..h).map(|_| self.s()).collect()
        }
    }
}