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 Lua (Lua 5.3.5) (judge verdict: Accepted)

local mod = 1000000007

local function bmul(x, y)
  return (x * y) % mod
end

local function badd(x, y)
  return (x + y) % mod
end

local function bsub(x, y)
  return x < y and x - y + mod or x - y
end

local function getgcd(x, y)
  while 0 < x do
    x, y = y % x, x
  end
  return y
end

local x1, y1 = {}, {}
local x2, y2 = {}, {}
local xzero = 0
local yzero = 0
local zero = 0

local n = io.read("*n", "*l")
for i = 1, n do
  local a, b = io.read("*n", "*n")
  if a == 0 and b == 0 then
    zero = zero + 1
  elseif a == 0 then
    xzero = xzero + 1
  elseif b == 0 then
    yzero = yzero + 1
  else
    if b < 0 then
      a, b = -a, -b
    end
    if 0 < a then
      local gcd = getgcd(a, b)
      table.insert(x1, a // gcd)
      table.insert(y1, b // gcd)
    else
      local gcd = getgcd(-a, b)
      table.insert(x2, -a // gcd)
      table.insert(y2, b // gcd)
    end
  end
end

local pow2 = {1}
for i = 1, 200010 do
  pow2[i + 1] = (pow2[i] * 2) % mod
end

local ret = badd(pow2[xzero + 1], pow2[yzero + 1])
ret = bsub(ret, 1)

local m1 = {}
for i1 = 1, #x1 do
  local xs, ys = x1[i1], y1[i1]
  if not m1[xs] then m1[xs] = {} end
  if not m1[xs][ys] then
    m1[xs][ys] = 1
  else
    m1[xs][ys] = m1[xs][ys] + 1
  end
end

local m2 = {}
for i2 = 1, #x2 do
  local xs, ys = x2[i2], y2[i2]
  if not m2[ys] then m2[ys] = {} end
  if not m2[ys][xs] then
    m2[ys][xs] = 1
  else
    m2[ys][xs] = m2[ys][xs] + 1
  end
end
for k1, v in pairs(m1) do
  if m2[k1] then
    local m2k1 = m2[k1]
    for k2, c1 in pairs(v) do
      local c2 = m2k1[k2]
      if c2 then
        ret = bmul(ret, bsub(badd(pow2[c1 + 1], pow2[c2 + 1]), 1))
        m2k1[k2] = nil
      else
        ret = bmul(ret, pow2[c1 + 1])
      end
    end
  else
    for k2, c1 in pairs(v) do
      ret = bmul(ret, pow2[c1 + 1])
    end
  end
end
for k1, v in pairs(m2) do
  for k2, c in pairs(v) do
    ret = bmul(ret, pow2[c + 1])
  end
end
ret = bsub(ret, 1)
ret = badd(ret, zero)
print(ret)