Problem:
Score: 600 points

Problem Statement

Takahashi and Aoki are training for long-distance races in an infinitely long straight course running from west to east.

They start simultaneously at the same point and moves as follows towards the east:

Takahashi runs A_1 meters per minute for the first T_1 minutes, then runs at A_2 meters per minute for the subsequent T_2 minutes, and alternates between these two modes forever.

Aoki runs B_1 meters per minute for the first T_1 minutes, then runs at B_2 meters per minute for the subsequent T_2 minutes, and alternates between these two modes forever.

How many times will Takahashi and Aoki meet each other, that is, come to the same point? We do not count the start of the run. If they meet infinitely many times, report that fact.

Constraints

1 \leq T_i \leq 100000

1 \leq A_i \leq 10^{10}

1 \leq B_i \leq 10^{10}

A_1 \neq B_1

A_2 \neq B_2

All values in input are integers.

Input

Input is given from Standard Input in the following format:

T_1 T_2
A_1 A_2
B_1 B_2

Output

Print the number of times Takahashi and Aoki will meet each other.

If they meet infinitely many times, print infinity instead.

Sample Input 1

1 2
10 10
12 4

Sample Output 1

1

They will meet just once, \frac{4}{3} minutes after they start, at \frac{40}{3} meters from where they start.

Sample Input 2

100 1
101 101
102 1

Sample Output 2

infinity

They will meet 101, 202, 303, 404, 505, 606, ... minutes after they start, that is, they will meet infinitely many times.

Sample Input 3

12000 15700
3390000000 3810000000
5550000000 2130000000

Sample Output 3

113

The values in input may not fit into a 32-bit integer type.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros, non_snake_case, bare_trait_objects)]
use std::cmp::Ordering::{self, Greater, Less};
use std::cmp::{max, min};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::{self, BufReader, BufWriter, Read, Write};

//{{{ macros missing in 1.15.1
#[macro_use]
mod macros {
    macro_rules! eprint {
        ($($t:tt)*) => {{
            use ::std::io::Write;
            let _ = write!(::std::io::stderr(), $($t)*);
        }};
    }
    macro_rules! eprintln {
        () => { eprintln!(""); };
        ($($t:tt)*) => {{
            use ::std::io::Write;
            let _ = writeln!(::std::io::stderr(), $($t)*);
        }};
    }
    macro_rules! dbg {
        ($val:expr) => {
            match $val {
                tmp => {
                    eprintln!(
                        "[{}:{}] {} = {:#?}",
                        file!(),
                        line!(),
                        stringify!($val),
                        &tmp
                    );
                    tmp
                }
            }
        };
    }
}
//}}}

fn solve_f<R: Read>(mut sc: Scanner<R>) -> isize {
    let t1: isize = sc.read();
    let t2: isize = sc.read();
    let a1: isize = sc.read();
    let a2: isize = sc.read();
    let b1: isize = sc.read();
    let b2: isize = sc.read();
    let (a1, a2, b1, b2) = if a1 < b1 {
        (a1, a2, b1, b2)
    } else {
        (b1, b2, a1, a2)
    };
    if a2 < b2 {
        return 0;
    }
    let ma = t1 * a1;
    let mb = t1 * b1;
    let sa = ma + t2 * a2;
    let sb = mb + t2 * b2;
    if sa == sb {
        return -1;
    }
    if sa < sb {
        return 0;
    }

    let mut ans = (mb - ma) / (sa - sb) * 2 + 1;
    if (mb - ma) % (sa - sb) == 0 {
        ans -= 1;
    }
    ans
}

fn solve<R: Read, W: Write>(mut sc: Scanner<R>, mut w: BufWriter<W>) {
    let ans = solve_f(sc);
    if ans != -1 {
        writeln!(w, "{}", ans);
    } else {
        writeln!(w, "infinity");
    }
}

fn run() {
    let stdin = io::stdin();
    let stdout = io::stdout();

    let input = stdin.lock();
    let output = BufWriter::new(stdout.lock());

    solve(Scanner::new(input), output);
}

fn main() {
    std::thread::Builder::new()
        .name("run".to_string())
        .stack_size(256 * 1024 * 1024)
        .spawn(run)
        .unwrap()
        .join()
        .unwrap()
}

//{{{ utils
pub struct Tokenizer<R: std::io::Read> {
    reader: R,
}

impl<R: std::io::Read> Tokenizer<R> {
    pub fn new(r: R) -> Tokenizer<R> {
        Tokenizer { reader: r }
    }
}

fn is_whitespace(b: u8) -> bool {
    b == b' ' || b == b'\n' || b == b'\r' || b == b'\t'
}

impl<R: std::io::Read> std::iter::Iterator for Tokenizer<R> {
    type Item = String;
    fn next(&mut self) -> Option<Self::Item> {
        let buf = std::io::Read::bytes(self.reader.by_ref())
            .map(|b| b.unwrap_or(b' '))
            .skip_while(|&b| is_whitespace(b))
            .take_while(|&b| !is_whitespace(b))
            .collect::<Vec<_>>();
        if buf.is_empty() {
            None
        } else {
            Some(unsafe { String::from_utf8_unchecked(buf) })
        }
    }
}

pub struct Scanner<R: std::io::Read> {
    tokenizer: Tokenizer<R>,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn new(r: R) -> Scanner<R> {
        Scanner {
            tokenizer: Tokenizer::new(r),
        }
    }
}

pub trait FromTokens {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self;
}

macro_rules! from_tokens_primitives {
    ($($t:ty),*) => { $(
        impl FromTokens for $t {
            fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
                tokens
                    .next()
                    .expect("no more token")
                    .parse()
                    .ok()
                    .expect("Parse error")
            }
        }
    )* }
}

// FIXME: Can it be implemented by FromStr?
from_tokens_primitives! {
    String,
    bool,
    f32,
    f64,
    isize,
    i8,
    i16,
    i32,
    i64,
    usize,
    u8,
    u16,
    u32,
    u64
}

impl<T1: FromTokens, T2: FromTokens> FromTokens for (T1, T2) {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (T1::from_tokens(tokens), T2::from_tokens(tokens))
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens> FromTokens for (T1, T2, T3) {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
        )
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens, T4: FromTokens> FromTokens
    for (T1, T2, T3, T4)
{
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
            T4::from_tokens(tokens),
        )
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens, T4: FromTokens, T5: FromTokens> FromTokens
    for (T1, T2, T3, T4, T5)
{
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
            T4::from_tokens(tokens),
            T5::from_tokens(tokens),
        )
    }
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: FromTokens>(&mut self) -> T {
        T::from_tokens(&mut self.tokenizer)
    }

    pub fn read_vec<T: FromTokens>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn read_chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}

pub trait SetMinMax {
    fn set_min(&mut self, v: Self) -> bool;
    fn set_max(&mut self, v: Self) -> bool;
}

impl<T> SetMinMax for T
where
    T: PartialOrd,
{
    fn set_min(&mut self, v: T) -> bool {
        *self > v && {
            *self = v;
            true
        }
    }
    fn set_max(&mut self, v: T) -> bool {
        *self < v && {
            *self = v;
            true
        }
    }
}

#[derive(PartialEq, Eq, Debug, Copy, Clone, Default, Hash)]
pub struct Reverse<T>(pub T);

impl<T: PartialOrd> PartialOrd for Reverse<T> {
    #[inline]
    fn partial_cmp(&self, other: &Reverse<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }

    #[inline]
    fn lt(&self, other: &Self) -> bool {
        other.0 < self.0
    }
    #[inline]
    fn le(&self, other: &Self) -> bool {
        other.0 <= self.0
    }
    #[inline]
    fn ge(&self, other: &Self) -> bool {
        other.0 >= self.0
    }
    #[inline]
    fn gt(&self, other: &Self) -> bool {
        other.0 > self.0
    }
}

impl<T: Ord> Ord for Reverse<T> {
    #[inline]
    fn cmp(&self, other: &Reverse<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}

#[derive(PartialEq, PartialOrd, Debug, Copy, Clone, Default)]
pub struct Num(pub f64);

impl Eq for Num {}

impl Ord for Num {
    fn cmp(&self, other: &Num) -> Ordering {
        self.0
            .partial_cmp(&other.0)
            .expect("unexpected NaN when compare")
    }
}

// See https://docs.rs/superslice/1.0.0/superslice/trait.Ext.html
pub trait SliceExt {
    type Item;

    fn lower_bound(&self, x: &Self::Item) -> usize
    where
        Self::Item: Ord;

    fn lower_bound_by<'a, F>(&'a self, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering;

    fn lower_bound_by_key<'a, K, F>(&'a self, k: &K, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord;

    fn upper_bound(&self, x: &Self::Item) -> usize
    where
        Self::Item: Ord;

    fn upper_bound_by<'a, F>(&'a self, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering;

    fn upper_bound_by_key<'a, K, F>(&'a self, k: &K, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord;
}

impl<T> SliceExt for [T] {
    type Item = T;

    fn lower_bound(&self, x: &Self::Item) -> usize
    where
        T: Ord,
    {
        self.lower_bound_by(|y| y.cmp(x))
    }

    fn lower_bound_by<'a, F>(&'a self, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering,
    {
        let s = self;
        let mut size = s.len();
        if size == 0 {
            return 0;
        }
        let mut base = 0usize;
        while size > 1 {
            let half = size / 2;
            let mid = base + half;
            let cmp = f(unsafe { s.get_unchecked(mid) });
            base = if cmp == Less { mid } else { base };
            size -= half;
        }
        let cmp = f(unsafe { s.get_unchecked(base) });
        base + (cmp == Less) as usize
    }

    fn lower_bound_by_key<'a, K, F>(&'a self, k: &K, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord,
    {
        self.lower_bound_by(|e| f(e).cmp(k))
    }

    fn upper_bound(&self, x: &Self::Item) -> usize
    where
        T: Ord,
    {
        self.upper_bound_by(|y| y.cmp(x))
    }

    fn upper_bound_by<'a, F>(&'a self, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering,
    {
        let s = self;
        let mut size = s.len();
        if size == 0 {
            return 0;
        }
        let mut base = 0usize;
        while size > 1 {
            let half = size / 2;
            let mid = base + half;
            let cmp = f(unsafe { s.get_unchecked(mid) });
            base = if cmp == Greater { base } else { mid };
            size -= half;
        }
        let cmp = f(unsafe { s.get_unchecked(base) });
        base + (cmp != Greater) as usize
    }

    fn upper_bound_by_key<'a, K, F>(&'a self, k: &K, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord,
    {
        self.upper_bound_by(|e| f(e).cmp(k))
    }
}

//}}}