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Distinct bytes used: X+B.
Try it online.
Explanation:
This is a subset of my answer for the Fewest (distinct) characters for Turing Completeness challenge.
X: push 1 to the stack +: adds the top two values on the stack together B: given two integer arguments, it'll convert the first into a base of the second
These three characters can be used to create any character of the 05AB1E codepage, including the 29 distinct characters we need for the output The quick brown fox jumps over the lazy dog.
In the challenge I've linked, we use an additional .V (evaluate as 05AB1E code) to open up all functionality. In this case however, we are only interested in printing the output. In the new version of 05AB1E, we could for example accomplish this by adding a ? after every B to print the individual characters; or we could add a trailing J to join all characters on the stack together, after which it'll be output implicitly.
In the legacy version of 05AB1E, which is built in Python (the new version is built in Elixir), + will also concat non-numeric strings, so we can add 42 trailing + to concat all characters on the stack together to a single string, after which it is output implicitly (with trailing newline).