The sensors have led you to the origin of the distress signal: yet another handheld device, just like the one the Elves gave you. However, you don't see any Elves around; instead, the device is surrounded by elephants! They must have gotten lost in these tunnels, and one of the elephants apparently figured out how to turn on the distress signal.
The ground rumbles again, much stronger this time. What kind of cave is this, exactly? You scan the cave with your handheld device; it reports mostly igneous rock, some ash, pockets of pressurized gas, magma... this isn't just a cave, it's a volcano!
You need to get the elephants out of here, quickly. Your device estimates that you have **30 minutes** before the volcano erupts, so you don't have time to go back out the way you came in.
You scan the cave for other options and discover a network of pipes and pressure-release **valves**. You aren't sure how such a system got into a volcano, but you don't have time to complain; your device produces a report (your puzzle input) of each valve's **flow rate** if it were opened (in pressure per minute) and the tunnels you could use to move between the valves.
There's even a valve in the room you and the elephants are currently standing in labeled `AA`. You estimate it will take you one minute to open a single valve and one minute to follow any tunnel from one valve to another. What is the most pressure you could release?
For example, suppose you had the following scan output:
All of the valves begin **closed**. You start at valve `AA`, but it must be damaged or jammed or something: its flow rate is `0`, so there's no point in opening it. However, you could spend one minute moving to valve `BB` and another minute opening it; doing so would release pressure during the remaining **28 minutes** at a flow rate of `13`, a total eventual pressure release of `28 * 13 = 364`. Then, you could spend your third minute moving to valve `CC` and your fourth minute opening it, providing an additional **26 minutes** of eventual pressure release at a flow rate of `2`, or **`52`** total pressure released by valve `CC`.
Making your way through the tunnels like this, you could probably open many or all of the valves by the time 30 minutes have elapsed. However, you need to release as much pressure as possible, so you'll need to be methodical. Instead, consider this approach:
You're worried that even with an optimal approach, the pressure released won't be enough. What if you got one of the elephants to help you?
It would take you 4 minutes to teach an elephant how to open the right valves in the right order, leaving you with only **26 minutes** to actually execute your plan. Would having two of you working together be better, even if it means having less time? (Assume that you teach the elephant before opening any valves yourself, giving you both the same full 26 minutes.)
In the example above, you could teach the elephant to help you as follows:
