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 C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using CompLib.Collections;
using CompLib.Mathematics;
using CompLib.Util;

public class Program
{
    int N;
    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        var plus = new HashMap<Iwashi, int>();
        var minus = new HashMap<Iwashi, int>();
        // a 0, 0 b
        int a0 = 0;
        int b0 = 0;
        int zero = 0;
        for (int i = 0; i < N; i++)
        {
            long a = sc.NextLong();
            long b = sc.NextLong();

            if (a != 0 && b == 0)
            {
                a0++;
                continue;
            }
            else if (a == 0 && b != 0)
            {
                b0++;
                continue;
            }
            else if (a == 0 && b == 0)
            {
                zero++;
                continue;
            }
            if (a < 0)
            {
                a = -a;
                b = -b;
            }

            long gcd = MathEx.GCD(Math.Abs(a), Math.Abs(b));
            var s = new Iwashi() { A = a / gcd, B = b / gcd };
            if (b > 0)
            {
                plus[s]++;
            }
            else
            {
                minus[s]++;
            }
        }
        ModInt[] pow2 = new ModInt[200001];
        pow2[0] = 1;
        for (int i = 1; i <= 200000; i++)
        {
            pow2[i] = pow2[i - 1] * 2;
        }
        ModInt ans = (pow2[a0] + pow2[b0] - 1);
        foreach (var p in plus)
        {
            var s = p.Key;
            Iwashi target;
            target = new Iwashi() { A = s.B, B = -s.A };
            int l = p.Value;
            int r = minus[target];
            var ll = pow2[l];
            var rr = pow2[r];
            ans *= (ll + rr - 1);
            minus.Remove(target);
        }

        foreach (var p in minus)
        {
            ans *= pow2[p.Value];
        }
        Console.WriteLine(ans - 1 + zero);
    }

    public static void Main(string[] args) => new Program().Solve();
}


namespace CompLib.Collections
{
    using System.Collections.Generic;
    public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
    {
        public new TValue this[TKey key]
        {
            get
            {
                TValue o;
                return TryGetValue(key, out o) ? o : default(TValue);
            }
            set { base[key] = value; }
        }
    }
}

// https://bitbucket.org/camypaper/complib
namespace CompLib.Mathematics
{
    using System;
    using System.Collections.Generic;
    #region GCD LCM
    /// <summary>
    /// 様々な数学的関数の静的メソッドを提供します．
    /// </summary>
    public static partial class MathEx
    {
        /// <summary>
        /// 2 つの整数の最大公約数を求めます．
        /// </summary>
        /// <param name="n">最初の値</param>
        /// <param name="m">2 番目の値</param>
        /// <returns>2 つの整数の最大公約数</returns>
        /// <remarks>ユークリッドの互除法に基づき最悪計算量 O(log N) で実行されます．</remarks>
        public static int GCD(int n, int m) { return (int)GCD((long)n, m); }


        /// <summary>
        /// 2 つの整数の最大公約数を求めます．
        /// </summary>
        /// <param name="n">最初の値</param>
        /// <param name="m">2 番目の値</param>
        /// <returns>2 つの整数の最大公約数</returns>
        /// <remarks>ユークリッドの互除法に基づき最悪計算量 O(log N) で実行されます．</remarks>
        public static long GCD(long n, long m)
        {
            n = Math.Abs(n);
            m = Math.Abs(m);
            while (n != 0)
            {
                m %= n;
                if (m == 0) return n;
                n %= m;
            }
            return m;
        }


        /// <summary>
        /// 2 つの整数の最小公倍数を求めます．
        /// </summary>
        /// <param name="n">最初の値</param>
        /// <param name="m">2 番目の値</param>
        /// <returns>2 つの整数の最小公倍数</returns>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public static long LCM(long n, long m) { return (n / GCD(n, m)) * m; }
    }
    #endregion
    #region PrimeSieve
    public static partial class MathEx
    {
        /// <summary>
        /// ある値までに素数表を構築します．
        /// </summary>
        /// <param name="max">最大の値</param>
        /// <param name="primes">素数のみを入れた数列が返される</param>
        /// <returns>0 から max までの素数表</returns>
        /// <remarks>エラトステネスの篩に基づき，最悪計算量 O(N loglog N) で実行されます．</remarks>
        public static bool[] Sieve(int max, List<int> primes = null)
        {
            var isPrime = new bool[max + 1];
            for (int i = 2; i < isPrime.Length; i++) isPrime[i] = true;
            for (int i = 2; i * i <= max; i++)
                if (!isPrime[i]) continue;
                else for (int j = i * i; j <= max; j += i) isPrime[j] = false;
            if (primes != null) for (int i = 0; i <= max; i++) if (isPrime[i]) primes.Add(i);

            return isPrime;
        }
    }
    #endregion
}

// https://bitbucket.org/camypaper/complib
namespace CompLib.Mathematics
{
    #region ModInt
    /// <summary>
    /// [0,<see cref="Mod"/>) までの値を取るような数
    /// </summary>
    public struct ModInt
    {
        /// <summary>
        /// 剰余を取る値．
        /// </summary>
        public const long Mod = (int)1e9 + 7;

        /// <summary>
        /// 実際の数値．
        /// </summary>
        public long num;
        /// <summary>
        /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
        /// </summary>
        /// <param name="n">インスタンスが持つ値</param>
        /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
        public ModInt(long n) { num = n; }
        /// <summary>
        /// このインスタンスの数値を文字列に変換します．
        /// </summary>
        /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
        public override string ToString() { return num.ToString(); }
        public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
        public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
        public static ModInt operator *(ModInt l, ModInt r) { return new ModInt(l.num * r.num % Mod); }
        public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(long v, long k)
        {
            long ret = 1;
            for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
                if ((k & 1) == 1) ret = ret * v % Mod;
            return new ModInt(ret);
        }
        /// <summary>
        /// 与えられた数の逆元を計算します．
        /// </summary>
        /// <param name="v">逆元を取る対象となる数</param>
        /// <returns>逆元となるような値</returns>
        /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
        public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
    }
    #endregion
    #region Binomial Coefficient
    public class BinomialCoefficient
    {
        public ModInt[] fact, ifact;
        public BinomialCoefficient(int n)
        {
            fact = new ModInt[n + 1];
            ifact = new ModInt[n + 1];
            fact[0] = 1;
            for (int i = 1; i <= n; i++)
                fact[i] = fact[i - 1] * i;
            ifact[n] = ModInt.Inverse(fact[n]);
            for (int i = n - 1; i >= 0; i--)
                ifact[i] = ifact[i + 1] * (i + 1);
            ifact[0] = ifact[1];
        }
        public ModInt this[int n, int r]
        {
            get
            {
                if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
                return fact[n] * ifact[n - r] * ifact[r];
            }
        }
        public ModInt RepeatedCombination(int n, int k)
        {
            if (k == 0) return 1;
            return this[n + k - 1, k];
        }
    }
    #endregion
}


struct Iwashi
{
    public long A, B;
}

namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}