## Back From Summer '19 P4: Wesley And Cake

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Points: 25 (partial)
Time limit: 1.0s
Memory limit: 64M

Author:
Problem type
A cake fit for a king ~wleung_bvg.

Everybody knows that cake comes in two shapes, circular or rectangular.
Everybody except for Wesley.

In his defence:

You wouldn't buy a rectangular shaped pizza would you?

Thus Wesley has only ever cut his cake in one way.

• Imagine a Cartesian grid over the center of the cake labelled at .
• Make cuts over the line formed by within the cake's boundaries.

With Wesley's birthday coming up, his friends have decided to play a little prank on him. They have purchased a square shaped cake with side length and would like to know the side length of the largest axis-aligned square obtainable that does not intersect with any cut. A square intersects a cut if there is a non-zero area of the square on both sides of the cut. This means that touching a cut does not count as an intersection.

Slope will be given as where .

#### Input Specification

The first line will contain two integers, and , half the length of the cake, and the number of cuts Wesley makes.

The next lines will each contain two integers, , the numerator and denominator of the slope of the line used to determine the cut. and are guaranteed to be coprime, and the pair is guaranteed to be unique.

#### Output Specification

Output two positive integers, and space-separated on one line. This means the side length of the largest obtainable axis-aligned square is . Note that and must be coprime.

#### Sample Input 1

1 2
-1 1
1 1

#### Sample Output 1

2 3

#### Explanation For Sample 1

The following image shows the cake:

The red square is the cake, and the blue and green lines shows the two cuts. The black line shows one of the many different largest squares obtainable.

#### Sample Input 2

2 2
-1 2
1 1

#### Sample Output 2

3 2

#### Sample Input 3

1000 2
-1000 1
1000 1

#### Sample Output 3

2000000 2001