part2

README

--- Part Two ---
Upon further analysis, it doesn't seem like any hailstones will naturally collide. It's up to you to fix that!

You find a rock on the ground nearby. While it seems extremely unlikely, if you throw it just right, you should be able to hit every hailstone in a single throw!

You can use the probably-magical winds to reach any integer position you like and to propel the rock at any integer velocity. Now including the Z axis in your calculations, if you throw the rock at time 0, where do you need to be so that the rock perfectly collides with every hailstone? Due to probably-magical inertia, the rock won't slow down or change direction when it collides with a hailstone.

In the example above, you can achieve this by moving to position 24, 13, 10 and throwing the rock at velocity -3, 1, 2. If you do this, you will hit every hailstone as follows:

Hailstone: 19, 13, 30 @ -2, 1, -2
Collision time: 5
Collision position: 9, 18, 20

Hailstone: 18, 19, 22 @ -1, -1, -2
Collision time: 3
Collision position: 15, 16, 16

Hailstone: 20, 25, 34 @ -2, -2, -4
Collision time: 4
Collision position: 12, 17, 18

Hailstone: 12, 31, 28 @ -1, -2, -1
Collision time: 6
Collision position: 6, 19, 22

Hailstone: 20, 19, 15 @ 1, -5, -3
Collision time: 1
Collision position: 21, 14, 12
Above, each hailstone is identified by its initial position and its velocity. Then, the time and position of that hailstone's collision with your rock are given.

After 1 nanosecond, the rock has exactly the same position as one of the hailstones, obliterating it into ice dust! Another hailstone is smashed to bits two nanoseconds after that. After a total of 6 nanoseconds, all of the hailstones have been destroyed.

So, at time 0, the rock needs to be at X position 24, Y position 13, and Z position 10. Adding these three coordinates together produces 47. (Don't add any coordinates from the rock's velocity.)

Determine the exact position and velocity the rock needs to have at time 0 so that it perfectly collides with every hailstone. What do you get if you add up the X, Y, and Z coordinates of that initial position?