Trichain: The Capital "A"

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Trichain is a puzzle type I invented. It is somewhat similar to Nurikabe, but it uses a chain of L-trominoes instead of a continuous wall.

Example puzzle with solution:

Rules:

1. Numbers must be white, and match the size of their white islands.
2. Each white island can only contain at most one number. Some white islands may have no numbers.
3. Black squares must be part of L-trominoes.
4. Black L-trominoes cannot be adjacent horizontally or vertically, but all of them must be connected diagonally. In other words, one standing on a black square must be able to visit any other black square via Chess King’s moves, stepping on black squares only.

Now, solve the following puzzle. The question marks indicate unknown numbers, which may or may not be distinct.