## --- Day 7: Laboratories ---

You thank the cephalopods for the help and exit the trash compactor, finding yourself in the familiar halls of a North Pole research wing.

Based on the large sign that says "teleporter hub", they seem to be researching teleportation; you can't help but try it for yourself and step onto the large yellow teleporter pad.

Suddenly, you find yourself in an unfamiliar room! The room has no doors; the only way out is the teleporter. Unfortunately, the teleporter seems to be leaking magic smoke.

Since this is a teleporter lab, there are lots of spare parts, manuals, and diagnostic equipment lying around. After connecting one of the diagnostic tools, it helpfully displays error code 0H-N0, which apparently means that there's an issue with one of the tachyon manifolds.

You quickly locate a diagram of the tachyon manifold (your puzzle input). A tachyon beam enters the manifold at the location marked S; tachyon beams always move downward. Tachyon beams pass freely through empty space (.). However, if a tachyon beam encounters a splitter (^), the beam is stopped; instead, a new tachyon beam continues from the immediate left and from the immediate right of the splitter.

For example:
```txt
.......S.......
...............
.......^.......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............
```
In this example, the incoming tachyon beam (|) extends downward from S until it reaches the first splitter:
```txt
.......S.......
.......|.......
.......^.......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............
```
At that point, the original beam stops, and two new beams are emitted from the splitter:
```txt
.......S.......
.......|.......
......|^|......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............
```
Those beams continue downward until they reach more splitters:
```txt
.......S.......
.......|.......
......|^|......
......|.|......
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............
```
At this point, the two splitters create a total of only three tachyon beams, since they are both dumping tachyons into the same place between them:
```txt
.......S.......
.......|.......
......|^|......
......|.|......
.....|^|^|.....
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............
```
This process continues until all of the tachyon beams reach a splitter or exit the manifold:
```txt
.......S.......
.......|.......
......|^|......
......|.|......
.....|^|^|.....
.....|.|.|.....
....|^|^|^|....
....|.|.|.|....
...|^|^|||^|...
...|.|.|||.|...
..|^|^|||^|^|..
..|.|.|||.|.|..
.|^|||^||.||^|.
.|.|||.||.||.|.
|^|^|^|^|^|||^|
|.|.|.|.|.|||.|
```
To repair the teleporter, you first need to understand the beam-splitting properties of the tachyon manifold. In this example, a tachyon beam is split a total of 21 times.

Analyze your manifold diagram. How many times will the beam be split?