"I run into the garden and jump into the water."
Translation:Befutok a kertbe és beugrom a vízbe.
Look at the 3rd person singular of a verb to determine its behavior. Ugrik and its various forms like beugrik, kiugrik, and so on fall into a special class of verbs which end in -ik and which (may) take -m instead of -k in the 1st person singular indefinite. Their first person definite and indefinite forms are identical.
Some of the most common examples include eszik, iszik, lakik, dolgozik, and látszik.
In practice, you will not be misunderstood if you use the -k forms instead, and even among Hungarians there is not total agreement about exactly which -ik verbs should use the special endings and how important this is. Some people are more prescriptive about it than others. On the whole, it is not one of the more logical aspects of Hungarian grammar.
How about "A kertbe futok be és a vízbe ugrom be"?
I guess it's meaning would be something like "It's the garden that I run into and the water that I jump into"? That at least seems like it could have a use somewhere. Or is the Hungarian here more or less explicitly a sequence of events (or for some other reason), so that the focus position can't be used this way?