>>542224593
Took me a while to realize that the numbers aren't levels but actually show how many of those engines Ray gives you at the start of the run you’ve leveled up to Lv. 3.
You level the engines by collecting the resonia cards of their respective types, the ones that have a pop-up checkmark saying "Already have Gear of the same style."
After that, you just have to do a bunch of runs with the different engines until you level all of them to at least Lv. 3. I recommend trying to get them to Lv. 4 in your first runs so you can make sure to get all the polychrome in the Gear Database
When you're done, switch the Investigation Strategy and repeat it aaaaaaaall over again