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)

require 'set'

def get_ints
  gets.split.map(&:to_i)
end

def get_int
  gets.chomp.to_i
end

class Hash
  def put_set(key, value)
    self[key] = Set.new if self[key].nil?
    self[key].add(value)
  end

  def put(key, value)
    self[key] = [] if self[key].nil?
    self[key].push(value)
  end
end

MOD = 1000000007

params = []

N = get_int

N.times do |num|
  a, b = get_ints
  params[num] = { taste: a, smell: b }
end

premise = {}

params.each.with_index do |p, i|
  params[(i+1)...].each.with_index do |op, j|
    if p[:taste] * op[:taste] + p[:smell] * op[:smell] == 0
      premise.put(i, j)
    end
  end
end

stock = premise.reduce(Set.new) do |prev, (key, values)|
  values.each do |v|
    prev.add([key, v])
  end
  prev
end

result_num = 2 ** N - 1
result = {}
result[2] = stock.dup
minus = true

loop do
  break if stock.length == 0

  stock.each do |arr|
    if minus
      result_num -= 2 ** (N - arr.length)
    else
      result_num += 2 ** (N - arr.length)
    end
  end

  res = stock.to_a
  stock = Set.new
  res.each.with_index do |r, i|
    res[i+1..].each do |x|
      x = (r | x).sort
      result.put_set(x.length, x)
      stock.add(x)
    end
  end

  minus = !minus
end

print result_num