@AdarshSharma is fundamentally correct, but I wanted to build on his answer.
When dealing with compound subjects you have an important rule:
"Or" separates, "and" combines
There is no case when "A and B" would not lead to the plural form of a verb. It always implies "at least two."
"Or," on the other hand, separates the subjects, meaning they're treated individually. Usually the phrase suggests only one of the two subjects will be important, but that's not always the case. Just because I tell you, "John or Luke will visit you," doesn't mean John and Luke can't show up.
The important point is that when using "or," you grammatically don't know which subject will be important. In this case, the plurality of the subject closest to the verb dictates which form the verb will take.
One last point
You may run across (or be tempted to use) complex compound sentences without the benefit of more words. Thus:
Jack and Jill or Dick and Jane care about you.
Can be evaluated as:
Jack and (Jill or Dick) and Jane care about you.
or:
(Jack and Jill) or (Dick and Jane) care about you.
In this particular example, all solutions are plural in nature and use the same verb form. But what happens when the subjects aren't evenly distributed across the conjunctions?
Jack and Jill or Jane cares about you.
Our two possiblities are:
Jack and (Jill or Jane) care about you.
(Jack and Jill) or Jane cares about you.
After a bit of research, it appears there are no rules governing the precedence of conjunctions, meaning we cannot grammatically resolve which verb to use. If you wrote this in a story, your editor would hand it back and ask you to reword the sentence — and rightly so.
However, in the technical world where rules are needed to resolve logical problems, the conjunction "and" takes precedence over the conjunction "or." This permits us to resolve the problem. As more and more of our educated youth grow up amidst an increasingly technical world, I expect they will become exposed to this idea (through experience, if not formally anywhere) because people like me with a technical background will simply default to using the conjunctions that way. I expect that, in the end, a rule of conjunctive precedence will be integrated into English grammar. It will be interesting to see (if I live long enough) whether my prophecy proves true.
Gratefully, you can linguistically create the parentheses I used above:
Jack and either Jill or Jane care about you.
The word "either" identifies the "Jill or Jane" group as more important than "and" by itself. This isn't the official way to use the "either...or" conjunction, but it works. Thank goodness!
Facinating... blooming fascinating.
Edit: My answer was modified because @EdwinAshworth brought up a valuable issue. After so many years working in technical industries, it never even crossed my mind that there might not be a priority of conjunctions. It's been decades since my last English class, but only hours since I last needed to use logical rules. Consequently, the use of conjunctive precedence simply makes sense — so much so that trying to impose another priority than that developed for logic literally sounds bad to my ear. And that started my search....
Son of a gun.
Therefore, Mr. Ashworth was correct that I was bold, indeed, to assert the "rule." The answer you see above has been adjusted to account for this.
My thanks, Mr. Ashworth, for pointing that out.
NP₁norNP₂ VPmeans bothNP₁andNP₂do notVP. In other words, it's plural.