The space near Jupiter is not a very safe place; you need to be careful of a [big distracting red spot](https://en.wikipedia.org/wiki/Great_Red_Spot), extreme [radiation](https://en.wikipedia.org/wiki/Magnetosphere_of_Jupiter), and a [whole lot of moons](https://en.wikipedia.org/wiki/Moons_of_Jupiter#List) swirling around. You decide to start by tracking the four largest moons: **Io, Europa, Ganymede, and Callisto**.
After a brief scan, you calculate the **position of each moon** (your puzzle input). You just need to **simulate their motion** so you can avoid them.
Each moon has a 3-dimensional position (`x`, `y`, and `z`) and a 3-dimensional velocity. The position of each moon is given in your scan; the `x`, `y`, and `z` velocity of each moon starts at `0`.
Simulate the motion of the moons in **time steps**. Within each time step, first update the velocity of every moon by applying **gravity**. Then, once all moons' velocities have been updated, update the position of every moon by applying **velocity**. Time progresses by one step once all of the positions are updated.
To apply **gravity**, consider every **pair** of moons. On each axis (`x`, `y`, and `z`), the velocity of each moon changes by **exactly +1 or -1** to pull the moons together. For example, if Ganymede has an `x` position of `3`, and Callisto has a `x` position of `5`, then Ganymede's `x` velocity **changes by `+1`** (because `5 > 3`) and Callisto's `x` velocity **changes by `-1`** (because `3 < 5`). However, if the positions on a given axis are the same, the velocity on that axis **does not change** for that pair of moons.
Once all gravity has been applied, apply **velocity**: simply add the velocity of each moon to its own position. For example, if Europa has a position of `x=1, y=2, z=3` and a velocity of `x=-2, y=0,z=3`, then its new position would be `x=-1, y=2, z=6`. This process does not modify the velocity of any moon.
For example, suppose your scan reveals the following positions:
```
<x=-1,y=0,z=2>
<x=2,y=-10,z=-7>
<x=4,y=-8,z=8>
<x=3,y=5,z=-1>
```
Simulating the motion of these moons would produce the following:
```
After 0 steps:
pos=<x=-1,y=0,z=2>, vel=<x=0,y=0,z=0>
pos=<x=2,y=-10,z=-7>, vel=<x=0,y=0,z=0>
pos=<x=4,y=-8,z=8>, vel=<x=0,y=0,z=0>
pos=<x=3,y=5,z=-1>, vel=<x=0,y=0,z=0>
After 1 step:
pos=<x=2,y=-1,z=1>, vel=<x=3,y=-1,z=-1>
pos=<x=3,y=-7,z=-4>, vel=<x=1,y=3,z=3>
pos=<x=1,y=-7,z=5>, vel=<x=-3,y=1,z=-3>
pos=<x=2,y=2,z=0>, vel=<x=-1,y=-3,z=1>
After 2 steps:
pos=<x=5,y=-3,z=-1>, vel=<x=3,y=-2,z=-2>
pos=<x=1,y=-2,z=2>, vel=<x=-2,y=5,z=6>
pos=<x=1,y=-4,z=-1>, vel=<x=0,y=3,z=-6>
pos=<x=1,y=-4,z=2>, vel=<x=-1,y=-6,z=2>
After 3 steps:
pos=<x=5,y=-6,z=-1>, vel=<x=0,y=-3,z=0>
pos=<x=0,y=0,z=6>, vel=<x=-1,y=2,z=4>
pos=<x=2,y=1,z=-5>, vel=<x=1,y=5,z=-4>
pos=<x=1,y=-8,z=2>, vel=<x=0,y=-4,z=0>
After 4 steps:
pos=<x=2,y=-8,z=0>, vel=<x=-3,y=-2,z=1>
pos=<x=2,y=1,z=7>, vel=<x=2,y=1,z=1>
pos=<x=2,y=3,z=-6>, vel=<x=0,y=2,z=-1>
pos=<x=2,y=-9,z=1>, vel=<x=1,y=-1,z=-1>
After 5 steps:
pos=<x=-1,y=-9,z=2>, vel=<x=-3,y=-1,z=2>
pos=<x=4,y=1,z=5>, vel=<x=2,y=0,z=-2>
pos=<x=2,y=2,z=-4>, vel=<x=0,y=-1,z=2>
pos=<x=3,y=-7,z=-1>, vel=<x=1,y=2,z=-2>
After 6 steps:
pos=<x=-1,y=-7,z=3>, vel=<x=0,y=2,z=1>
pos=<x=3,y=0,z=0>, vel=<x=-1,y=-1,z=-5>
pos=<x=3,y=-2,z=1>, vel=<x=1,y=-4,z=5>
pos=<x=3,y=-4,z=-2>, vel=<x=0,y=3,z=-1>
After 7 steps:
pos=<x=2,y=-2,z=1>, vel=<x=3,y=5,z=-2>
pos=<x=1,y=-4,z=-4>, vel=<x=-2,y=-4,z=-4>
pos=<x=3,y=-7,z=5>, vel=<x=0,y=-5,z=4>
pos=<x=2,y=0,z=0>, vel=<x=-1,y=4,z=2>
After 8 steps:
pos=<x=5,y=2,z=-2>, vel=<x=3,y=4,z=-3>
pos=<x=2,y=-7,z=-5>, vel=<x=1,y=-3,z=-1>
pos=<x=0,y=-9,z=6>, vel=<x=-3,y=-2,z=1>
pos=<x=1,y=1,z=3>, vel=<x=-1,y=1,z=3>
After 9 steps:
pos=<x=5,y=3,z=-4>, vel=<x=0,y=1,z=-2>
pos=<x=2,y=-9,z=-3>, vel=<x=0,y=-2,z=2>
pos=<x=0,y=-8,z=4>, vel=<x=0,y=1,z=-2>
pos=<x=1,y=1,z=5>, vel=<x=0,y=0,z=2>
After 10 steps:
pos=<x=2,y=1,z=-3>, vel=<x=-3,y=-2,z=1>
pos=<x=1,y=-8,z=0>, vel=<x=-1,y=1,z=3>
pos=<x=3,y=-6,z=1>, vel=<x=3,y=2,z=-3>
pos=<x=2,y=0,z=4>, vel=<x=1,y=-1,z=-1>
```
Then, it might help to calculate the **total energy in the system**. The total energy for a single moon is its **potential energy** multiplied by its **kinetic energy**. A moon's **potential energy** is the sum of the [absolute values](https://en.wikipedia.org/wiki/Absolute_value) of its `x`, `y`, and `z` position coordinates. A moon's **kinetic energy** is the sum of the absolute values of its velocity coordinates. Below, each line shows the calculations for a moon's potential energy (`pot`), kinetic energy (`kin`), and total energy:
**What is the total energy in the system** after simulating the moons given in your scan for `1000` steps?
## Part 2
All this drifting around in space makes you wonder about the nature of the universe. Does history really repeat itself? You're curious whether the moons will ever return to a previous state.
Determine the **number of steps** that must occur before all of the moons' **positions and velocities** exactly match a previous point in time.
For example, the first example above takes 2772 steps before they exactly match a previous point in time; it eventually returns to the initial state:
```
After 0 steps:
pos=<x=-1,y=0,z=2>, vel=<x=0,y=0,z=0>
pos=<x=2,y=-10,z=-7>, vel=<x=0,y=0,z=0>
pos=<x=4,y=-8,z=8>, vel=<x=0,y=0,z=0>
pos=<x=3,y=5,z=-1>, vel=<x=0,y=0,z=0>
After 2770 steps:
pos=<x=2,y=-1,z=1>, vel=<x=-3,y=2,z=2>
pos=<x=3,y=-7,z=-4>, vel=<x=2,y=-5,z=-6>
pos=<x=1,y=-7,z=5>, vel=<x=0,y=-3,z=6>
pos=<x=2,y=2,z=0>, vel=<x=1,y=6,z=-2>
After 2771 steps:
pos=<x=-1,y=0,z=2>, vel=<x=-3,y=1,z=1>
pos=<x=2,y=-10,z=-7>, vel=<x=-1,y=-3,z=-3>
pos=<x=4,y=-8,z=8>, vel=<x=3,y=-1,z=3>
pos=<x=3,y=5,z=-1>, vel=<x=1,y=3,z=-1>
After 2772 steps:
pos=<x=-1,y=0,z=2>, vel=<x=0,y=0,z=0>
pos=<x=2,y=-10,z=-7>, vel=<x=0,y=0,z=0>
pos=<x=4,y=-8,z=8>, vel=<x=0,y=0,z=0>
pos=<x=3,y=5,z=-1>, vel=<x=0,y=0,z=0>
```
Of course, the universe might last for a **very long time** before repeating. Here's a copy of the second example from above:
```
<x=-8,y=-10,z=0>
<x=5,y=5,z=10>
<x=2,y=-7,z=3>
<x=9,y=-8,z=-3>
```
This set of initial positions takes `4686774924` steps before it repeats a previous state! Clearly, you might need to **find a more efficient way to simulate the universe**.
**How many steps does it take** to reach the first state that exactly matches a previous state?
