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: Time Limit Exceeded)

class Sardine
  def initialize(taste, fragrance)
    @taste = taste
    @fragrance = fragrance
  end

  attr_reader :taste, :fragrance

  def to_a
    [taste, fragrance]
  end

  def zero_vector?
    to_a.all?(&:zero?)
  end

  def vertical?
    to_a.first.zero?
  end

  def horizontal?
    to_a.last.zero?
  end

  def to_rational
    return :zero_vector if zero_vector?
    return :vertical if vertical?
    return :horizontal if horizontal?

    Rational(fragrance, taste)
  end
end

class ModPower
  def initialize(base, mod = nil)
    @base = base
    @mod = mod

    @h = {
      0 => 1,
      1 => (mod ? base % mod : base )
    }
  end

  attr_reader :base, :mod

  def get(n)
    if @h.key?(n)
      # puts "n: #{n}"
      return @h[n]
    end

    if n % 2 == 0
      v = get(n / 2) ** 2
      v %= mod if mod
    else
      v = (get(n / 2) ** 2) * get(1)
      v %= mod if mod
    end

    @h[n] = v
  end
end

n = gets.chomp.to_i
sardines = Array.new(n) { a, b = gets.chomp.split(/ /).map(&:to_i); Sardine.new(a, b) }

sardines_with_zero_vector, sardines_with_nonzero_vector = sardines.partition(&:zero_vector?)

sardine_groups = {}
sardines_with_nonzero_vector.each do |sardine|
  r = sardine.to_rational
  case r
  when :horizontal
    sardine_groups[:on_coordinate] ||= [[], []]
    sardine_groups[:on_coordinate][0].push(sardine)
  when :vertical
    sardine_groups[:on_coordinate] ||= [[], []]
    sardine_groups[:on_coordinate][1].push(sardine)
  when Proc.new(&:positive?)
    sardine_groups[r] ||= [[], []]
    sardine_groups[r][0].push(sardine)
  else
    sardine_groups[- (r ** -1)] ||= [[], []]
    sardine_groups[- (r ** -1)][1].push(sardine)
  end
end

mod = 1_000_000_007

def calc_sardine_groups(sardine_group, power_instance, mod)
  group_plus, group_minus = sardine_group
  j = power_instance.get(group_plus.size)
  k = power_instance.get(group_minus.size)
  (j + k - 1) % mod
end

i = 0
unless sardine_groups.empty?
  j = 1
  power_instance = ModPower.new(2, mod)
  sardine_groups.each do |key, sardine_group|
    j *= calc_sardine_groups(sardine_group, power_instance, mod) % mod
  end
  i += (j - 1) % mod
end

unless sardines_with_zero_vector.empty?
  i += sardines_with_zero_vector.size % mod
end

puts i % mod