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 Kotlin (1.3.71) (judge verdict: Wrong Answer)

import kotlin.math.pow

val M = 1000000007L

fun main() {
    val n = readLine()!!.toInt()
    var zeros = 0L
    val iwashiMap = mutableMapOf<Fraction, Int>()
    repeat(n) {
        val (a, b) = readLine()!!.split(" ")
        val aa = a.toLong()
        val bb = b.toLong()
        val f = fraction(aa, bb)
        if (f is Fraction.Zero) {
            zeros += 1
        } else {
            val v = iwashiMap.getOrDefault(f, 0)
            iwashiMap[f] = v + 1
        }
    }

    iwashiMap.keys.forEach { f ->
        val o = f.opposite()
        if (iwashiMap.containsKey(o)) f.hasOpposite = o
    }

    val resultMap = mutableMapOf<Set<Fraction>, Long>()

    iwashiMap.forEach { (k, v) -> resultMap[setOf(k)] = pow2(v) }
    for (size in 1 until iwashiMap.keys.size) {
        val candidates = resultMap.filterKeys { k -> k.size == size }
        candidates.forEach { (k, v) ->
            val candidateOpposites = k.mapNotNullTo(mutableSetOf()) { it.hasOpposite }
            iwashiMap.keys.forEach inner@{ f ->
                if (candidateOpposites.contains(f)) return@inner
                if (k.contains(f)) return@inner
                resultMap[k + f] = v * pow2(iwashiMap.getValue(f)) % M
            }
        }
    }
    val index = resultMap.values.fold(0L) { b, a -> (a + b) % M }
    val result = (zeros + index) % M
    println(result)
}

fun pow2(n: Int): Long = (2.0.pow(n).toLong() - 1) % M

sealed class Fraction {
    data class F(val m: Long, val n: Long) : Fraction() {
        override fun opposite(): Fraction = if (m > 0) F(-n, m) else F(n, -m)
    }

    object ZeroOne : Fraction() {
        override fun opposite(): Fraction = OneZero
    }

    object OneZero : Fraction() {
        override fun opposite(): Fraction = ZeroOne
    }

    object Zero : Fraction() {
        override fun opposite(): Fraction = Zero
    }

    abstract fun opposite(): Fraction
    var hasOpposite: Fraction? = null
}

fun fraction(m: Long, n: Long): Fraction {
    return when {
        m == 0L && n == 0L -> Fraction.Zero
        m == 0L -> Fraction.ZeroOne
        n == 0L -> Fraction.OneZero
        else -> {
            val a = gcd(m, n)
            Fraction.F(m / a, n / a)
        }
    }
}

tailrec fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b)