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 Ruby (2.7.1) (judge verdict: Accepted)

# この問題の解説ブログはこちら https://ysk-pro.hatenablog.com/entry/abc168-e-bullet

MOD = 1000000007
N = gets.chomp.to_i

group = Hash.new(0)
worst_iwashi_count = 0 # 最悪のイワシを管理する変数

N.times do |i|
  a, b = gets.chomp.split.map(&:to_i)

  if a == 0 && b == 0 # 最悪のイワシのケース
    worst_iwashi_count += 1
    next
  end

  if a == 0 # A = 0 のイワシのケース
    group[[0, -1]] += 1
    next
  end

  if b == 0 # A = 0 のイワシのケース
    group[[1, 0]] += 1
    next
  end

  # 最大公約数でそれぞれを割る
  gcd = a.gcd(b)
  a /= gcd
  b /= gcd

  # 正負を揃える
  if a < 0
    a *= -1
    b *= -1
  end

  group[[a, b]] += 1
end

ans = 1

group.each do |(a, b), count|
  # ペアとなるグループに含まれるイワシの数を数える
  # ペアとして計算されたグループは、重複して計算されることを防ぐために削除する
  if b < 0
    pair_count = group[[-b, a]]
    group.delete([-b, a])
  else
    pair_count = group[[b, -a]]
    group.delete([b, -a])
  end

  ans *= (2 ** count - 1) + (2 ** pair_count - 1) + 1
  ans %= MOD
end

p (ans + worst_iwashi_count - 1) % MOD