There are various ways, the most common are to simply change the subject, use passive voice, or, uh, do the latter part of the example. I'm not good with theoretical side of grammar.
For example "The <action> was completed swiftly, allowing me to move on to the next problem."
Ocassionally you can also skip the person entirely, for example "I started sorting the scrap; this went there, that went there, and everything that was too damaged was thrown into a container to take out later." - there is only one "I" where you could easily fit five, but the context makes it easy to see who the person doing things is.
Generally, don't replace the "I", but think up ways to say the same thing in a different way; as long as it's not too convoluted it will work fine.
And sometimes when you've no idea what to do to to make it sound better, you can always add or remove a short sentence to increase the spacing between individual "I".