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PUZZLE #5: WHAT'S THE NEXT NUMBER?

Here are the first few entries of an interesting infinite sequence of
numbers:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
The problem is to determine a believable algorithm for generating
the rest of the numbers in the sequence, and to specify exactly
which (base ten) digits can appear.
(solution)