Problem:
Score: 500 points

Problem Statement

We have caught N sardines. The deliciousness and fragrantness of the i-th sardine is A_i and B_i, respectively.

We will choose one or more of these sardines and put them into a cooler. However, two sardines on bad terms cannot be chosen at the same time.

The i-th and j-th sardines (i \neq j) are on bad terms if and only if A_i \cdot A_j + B_i \cdot B_j = 0.

In how many ways can we choose the set of sardines to put into the cooler? Since the count can be enormous, print it modulo 1000000007.

Constraints

All values in input are integers.

1 \leq N \leq 2 \times 10^5

-10^{18} \leq A_i, B_i \leq 10^{18}

Input

Input is given from Standard Input in the following format:

N
A_1 B_1
:
A_N B_N

Output

Print the count modulo 1000000007.

Sample Input 1

3
1 2
-1 1
2 -1

Sample Output 1

5

There are five ways to choose the set of sardines, as follows:

The 1-st

The 1-st and 2-nd

The 2-nd

The 2-nd and 3-rd

The 3-rd

Sample Input 2

10
3 2
3 2
-1 1
2 -1
-3 -9
-8 12
7 7
8 1
8 2
8 4

Sample Output 2

479

Solution in Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);
impl ModInt {
    pub const MOD: u32 = 1_000_000_007;
    pub fn inv(self) -> Self {
        if self.0 == 0 {
            panic!();
        }
        self.pow(Self::MOD - 2)
    }
    pub fn one() -> Self {
        Self(1)
    }
    pub fn pow(self, e: u32) -> Self {
        if e == 0 {
            return Self::new(1);
        }
        let mut res = self.pow(e >> 1);
        res *= res;
        if e & 1 == 1 {
            res *= self;
        }
        res
    }
    pub fn zero() -> Self {
        Self(0)
    }
    fn new(n: i64) -> Self {
        let m = Self::MOD as i64;
        let mut n = n % m;
        if n.is_negative() {
            n += m;
        }
        Self(n as u32)
    }
}
impl From<i32> for ModInt {
    fn from(n: i32) -> Self {
        ModInt::from(n as i64)
    }
}
impl From<i64> for ModInt {
    fn from(n: i64) -> Self {
        Self::new(n)
    }
}
impl From<isize> for ModInt {
    fn from(n: isize) -> Self {
        ModInt::from(n as i64)
    }
}
impl From<u32> for ModInt {
    fn from(n: u32) -> Self {
        ModInt::from(n as u64)
    }
}
impl From<u64> for ModInt {
    fn from(n: u64) -> Self {
        Self::new(n as i64)
    }
}
impl From<usize> for ModInt {
    fn from(n: usize) -> Self {
        ModInt::from(n as u64)
    }
}
impl Into<i32> for ModInt {
    fn into(self) -> i32 {
        self.0 as i32
    }
}
impl Into<i64> for ModInt {
    fn into(self) -> i64 {
        self.0 as i64
    }
}
impl Into<isize> for ModInt {
    fn into(self) -> isize {
        self.0 as isize
    }
}
impl Into<u32> for ModInt {
    fn into(self) -> u32 {
        self.0
    }
}
impl Into<u64> for ModInt {
    fn into(self) -> u64 {
        self.0 as u64
    }
}
impl Into<usize> for ModInt {
    fn into(self) -> usize {
        self.0 as usize
    }
}
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        Self::new((self.0 + rhs.0) as i64)
    }
}
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        self * rhs.inv()
    }
}
impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) * (rhs.0 as i64))
    }
}
impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) - (rhs.0 as i64))
    }
}
impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

fn normalize(pair: (i64, i64)) -> (i64, i64) {
    let (a, b) = pair;
    if a == 0 && b == 0 {
        return (a, b);
    }
    let gcd = a.gcd(&b);
    let a = a / gcd;
    let b = b / gcd;
    if a == 0 || b == 0 {
        let a = if a != 0 { 1 } else { a };
        let b = if b != 0 { 1 } else { b };
        return (a, b);
    }
    let half_negative = a.is_negative() && !b.is_negative() || !a.is_negative() && b.is_negative();
    if half_negative {
        let a = a.abs();
        let b = b.abs();
        (a * -1, b)
    } else {
        (a.abs(), b.abs())
    }
}

fn main() {
    input! {
        n: usize,
        abv: [(i64, i64); n]
    };

    let mut counts = HashMap::new();
    for ab in abv {
        let pair = normalize(ab);
        *counts.entry(pair).or_insert(0_usize) += 1;
    }
    let mut res = ModInt::one();
    let zero_count = ModInt::from(*counts.get(&(0, 0)).unwrap_or(&0));
    let keys = counts.keys().copied().collect::<Vec<_>>();
    for k in keys {
        if !counts.contains_key(&k) {
            continue;
        }

        if k == (0, 0) {
            continue;
        }

        let (a, b) = k;
        let mut count = ModInt::zero();
        let c1 = *counts.get(&k).unwrap();
        count += ModInt::from(2).pow(c1 as u32) - ModInt::one();
        let k2 = normalize((b, -a));
        match counts.get(&k2) {
            Some(&c2) => {
                count += ModInt::from(2).pow(c2 as u32) - ModInt::one();
            }
            None => {}
        };
        count += ModInt::one();
        res *= count;

        counts.remove(&k);
        counts.remove(&k2);
    }
    res += zero_count;
    res -= ModInt::one();
    println!("{}", res);
}