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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;

class Solver
{
    public Scanner sc;
    public void Solve()
    {
        var N = sc.Int;
        var zero = 0;
        long c = 0, d = 0;
        var ct = new Dictionary<Frac, int>();
        for(int i = 0; i < N; i++)
        {
            long a, b;
            sc.ArrLong.Out(out a, out b);
            if (a == 0 && b == 0) zero++;
            else if (a == 0) d++;
            else if (b == 0) c++;
            else
            {
                Frac f = (a, b);
                if (!ct.ContainsKey(f)) ct[f] = 0;
                ct[f]++;
            }
        }
        ModInt res = ModInt.Pow(2,c) + ModInt.Pow(2,d) - 1;
        var use = new HashSet<Frac>();
        foreach(var f in ct.Keys)
        {
            var l = ModInt.Pow(2, ct[f]);
            Frac p = (-f.s, f.f);
            if (use.Contains(p)) continue;
            if (ct.ContainsKey(p)) { l += ModInt.Pow(2, ct[p]) - 1; }
            res *= l;use.Add(f);
        }
        Console.WriteLine(res-1+zero);
    }
}

class Frac
{
    public long f, s;
    public void Normalize()
    {
        if (s < 0) { f = -f; s = -s; }
        var g = GCD(Abs(f), Abs(s));
        if (g == 0) { f = 0; s = 1; }
        else { f /= g; s /= g; }
    }
    public Frac(long f = 0, long s = 1) { this.f = f; this.s = s; Normalize(); }
    public double ToDouble() => (double)f / s;
    public override string ToString() => f + "/" + s;
    public static bool operator ==(Frac a, Frac b)
    {
        return a.f * b.s == a.s * b.f;
    }
    public static bool operator !=(Frac a, Frac b) => !(a == b);
    public static Frac operator +(Frac a, Frac b)
    {
        Frac rt = new Frac();
        rt.f = a.f * b.s + a.s * b.f;
        rt.s = a.s * b.s;
        rt.Normalize();
        return rt;
    }
    public static Frac operator -(Frac a, Frac b)
    {
        Frac rt = new Frac();
        rt.f = a.f * b.s - a.s * b.f;
        rt.s = a.s * b.s;
        rt.Normalize();
        return rt;
    }
    public static Frac operator *(Frac a, Frac b)
    {
        Frac rt = new Frac();
        rt.f = a.f * b.f;
        rt.s = a.s * b.s;
        rt.Normalize();
        return rt;
    }
    public static Frac operator /(Frac a, Frac b)
    {
        Frac rt = new Frac();
        rt.f = a.f * b.s;
        rt.s = a.s * b.f;
        rt.Normalize();
        return rt;
    }
    public static implicit operator Frac(long a) => new Frac(a);
    public static implicit operator Frac((long f, long s) p) => new Frac(p.f, p.s);
    public override bool Equals(object b) { if (b == null) return false;var w = (Frac)b;return w.f == f && w.s == s; }
    public override int GetHashCode()
    {
        unchecked
        {
            return (f.GetHashCode() << 5 + f.GetHashCode()) ^ s.GetHashCode();
        }
    }
    public long GCD(long num1, long num2)
        => num1 < num2 ? GCD(num2, num1) :
           num2 > 0 ? GCD(num2, num1 % num2) : num1;
}

public struct ModInt
{
    public const long MOD = (int)1e9 + 7;
    //public const long MOD = 998244353;
    public long Value { get; set; }
    public ModInt(long n = 0) { Value = n; }
    private static ModInt[] fac, inv, facinv;
    public override string ToString() => Value.ToString();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt l, ModInt r)
    {
        l.Value += r.Value;
        if (l.Value >= MOD) l.Value -= MOD;
        return l;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt l, ModInt r)
    {
        l.Value -= r.Value;
        if (l.Value < 0) l.Value += MOD;
        return l;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator *(ModInt l, ModInt r) => new ModInt(l.Value * r.Value % MOD);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator /(ModInt l, ModInt r) => l * Pow(r, MOD - 2);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator long(ModInt l) => l.Value;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator ModInt(long n)
    {
        n %= MOD; if (n < 0) n += MOD;
        return new ModInt(n);
    }

    public static ModInt Pow(ModInt m, long n)
    {
        if (n == 0) return 1;
        if (n % 2 == 0) return Pow(m * m, n >> 1);
        else return Pow(m * m, n >> 1) * m;
    }

    public static void Build(int n)
    {
        fac = new ModInt[n + 1];
        facinv = new ModInt[n + 1];
        inv = new ModInt[n + 1];
        inv[1] = fac[0] = fac[1] = facinv[0] = facinv[1] = 1;
        for (var i = 2; i <= n; i++)
        {
            fac[i] = fac[i - 1] * i;
            inv[i] = MOD - inv[MOD % i] * (MOD / i);
            facinv[i] = facinv[i - 1] * inv[i];
        }
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt Fac(int n) => fac[n];
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt Inv(int n) => inv[n];
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt FacInv(int n) => facinv[n];
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt Comb(int n, int r)
    {
        if (n < r) return 0;
        if (n == r) return 1;
        return fac[n] * facinv[r] * facinv[n - r];
    }
    public static ModInt Perm(int n, int r)
    {
        if (n < r) return 0;
        return fac[n] * facinv[n - r];
    }

}
#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        var sol = new Solver() { sc = new Scanner() };
        //var th = new Thread(sol.Solve, 1 << 26);th.Start();th.Join();
        sol.Solve();
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) > 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) < 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Out<T>(this IList<T> A, out T a) => a = A[0];
    public static void Out<T>(this IList<T> A,out T a,out T b) { a = A[0];b = A[1]; }
    public static void Out<T>(this IList<T> A,out T a,out T b,out T c) { A.Out(out a, out b);c = A[2]; }
    public static void Out<T>(this IList<T> A,out T a,out T b,out T c,out T d) { A.Out(out a, out b, out c);d = A[3]; }
    public static string Concat<T>(this IEnumerable<T> A, string sp) => string.Join(sp, A);
    public static IEnumerable<T> Shuffle<T>(this IEnumerable<T> A) => A.OrderBy(v => Guid.NewGuid());
    public static int CompareTo<T>(this T[] A, T[] B, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; for (var i = 0; i < Min(A.Length, B.Length); i++) { int c = cmp(A[i], B[i]); if (c > 0) return 1; else if (c < 0) return -1; } if (A.Length == B.Length) return 0; if (A.Length > B.Length) return 1; else return -1; }
    public static int ArgMax<T>(this IList<T> A, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; T max = A[0]; int rt = 0; for (int i = 1; i < A.Count; i++) if (cmp(max, A[i]) < 0) { max = A[i]; rt = i; } return rt; }
    public static T PopBack<T>(this List<T> A) { var v = A[A.Count - 1]; A.RemoveAt(A.Count - 1); return v; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    public void Make<T1, T2, T3, T4, T5, T6, T7>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6, out T7 v7) { Make(out v1, out v2, out v3, out v4, out v5, out v6); v7 = Next<T7>(); }
    public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}

#endregion