## --- Day 20: Particle Swarm ---

Suddenly, the GPU contacts you, asking for help. Someone has asked it to simulate *too many particles*, and it won't be able to finish them all in time to render the next frame at this rate.

It transmits to you a buffer (your puzzle input) listing each particle in order (starting with particle `0`

, then particle `1`

, particle `2`

, and so on). For each particle, it provides the `X`

, `Y`

, and `Z`

coordinates for the particle's position (`p`

), velocity (`v`

), and acceleration (`a`

), each in the format `<X,Y,Z>`

.

Each tick, all particles are updated simultaneously. A particle's properties are updated in the following order:

- Increase the
`X`

velocity by the`X`

acceleration. - Increase the
`Y`

velocity by the`Y`

acceleration. - Increase the
`Z`

velocity by the`Z`

acceleration. - Increase the
`X`

position by the`X`

velocity. - Increase the
`Y`

position by the`Y`

velocity. - Increase the
`Z`

position by the`Z`

velocity.

Because of seemingly tenuous rationale involving z-buffering, the GPU would like to know which particle will stay closest to position `<0,0,0>`

in the long term. Measure this using the Manhattan distance, which in this situation is simply the sum of the absolute values of a particle's `X`

, `Y`

, and `Z`

position.

For example, suppose you are only given two particles, both of which stay entirely on the X-axis (for simplicity). Drawing the current states of particles `0`

and `1`

(in that order) with an adjacent a number line and diagram of current `X`

positions (marked in parenthesis), the following would take place:

```
p=< 3,0,0>, v=< 2,0,0>, a=<-1,0,0> -4 -3 -2 -1 0 1 2 3 4
p=< 4,0,0>, v=< 0,0,0>, a=<-2,0,0> (0)(1)
p=< 4,0,0>, v=< 1,0,0>, a=<-1,0,0> -4 -3 -2 -1 0 1 2 3 4
p=< 2,0,0>, v=<-2,0,0>, a=<-2,0,0> (1) (0)
p=< 4,0,0>, v=< 0,0,0>, a=<-1,0,0> -4 -3 -2 -1 0 1 2 3 4
p=<-2,0,0>, v=<-4,0,0>, a=<-2,0,0> (1) (0)
p=< 3,0,0>, v=<-1,0,0>, a=<-1,0,0> -4 -3 -2 -1 0 1 2 3 4
p=<-8,0,0>, v=<-6,0,0>, a=<-2,0,0> (0)
```

At this point, particle `1`

will never be closer to `<0,0,0>`

than particle `0`

, and so, in the long run, particle `0`

will stay closest.

*Which particle will stay closest to position <0,0,0>* in the long term?