Hooks 11

The 5-by-5 grid below must be split into 5 L-shaped "hooks". A hook of size k is an L whose two arms are each k squares long (sharing the corner square): the largest is 5-by-5 (9 squares), the next is 4-by-4 (7 squares), then 3-by-3 (5 squares), 2-by-2 (3 squares), and the smallest is a single square. (The 9-square hook must run along two edges of the grid, the 7-square hook along two edges of the 4-by-4 square that is left, and so on, so the hooks nest inside one another.) Find where the hooks are. Then choose one hook to hold five 5's, another to hold four 4's, and so on down to one hook holding a single 1: every filled square of a hook holds that hook's digit, and a hook holding digit d has exactly d filled squares. The digit a hook holds does not have to equal its size. Squares in a hook that hold no digit stay empty.

Clues outside the grid: top of column c0 = 4, top of column c2 = 1, bottom of column c1 = 2, left of row r0 = Y, right of row r1 = N, left of row r3 = 3, right of row r4 = V.

Rules.

1. The filled squares must form one connected region. (Squares are connected if they are orthogonally adjacent.)
2. Every 2-by-2 block must contain at least one empty square.
3. The 15 filled squares must split into 3 different pentominoes (no shape repeated, counting rotations and reflections as the same shape). The digits in each pentomino must add up to a multiple of 5.
4. A clue outside the grid gives the first filled square you meet when looking into that row or column from that side. Empty squares are skipped. A number clue gives that square's digit. A letter clue names the pentomino that contains that square, using the standard names F, I, L, N, P, T, U, V, W, X, Y, Z.

The solution is unique.

Task. Find the orthogonally connected groups of empty squares in the completed grid and multiply their areas together.

Answer format: a single positive integer (the product of the areas of the orthogonally connected groups of empty squares; a lone empty square is a group of area 1).