Hooks #6

The 6-by-6 grid below can be split into six L-shaped hooks, one inside the next:

• The 6-hook is one full edge row of the grid together with one full edge column that meets it at a corner. It has 11 cells.
• Take the 6-hook away and a 5-by-5 square is left. The 5-hook is one full edge row and one full edge column of that square, meeting at a corner. It has 9 cells.
• Keep going. The 4-hook has 7 cells, the 3-hook has 5 cells and the 2-hook has 3 cells. The 1-hook is the single cell that remains.

At each step, the hook may use any of the four corners of the square that remains.

Find where the hooks go, then fill in digits:

• Put six 6's in the 6-hook, five 5's in the 5-hook, and so on, down to one 1 in the 1-hook. Every other cell stays empty.
• The filled cells must form one connected region. Cells count as connected if they share a side.
• Every 2-by-2 block of cells must contain at least one empty cell.

Clues. In a row or column, each maximal run of consecutive filled cells is read as one number, starting from the side where the clue is written. For example, the row 3 3 . 4 1 . reads as the numbers 33 and 41 from the left, and as 14 and 33 from the right.

• A plain number is the product of the numbers in that line.
• A number in parentheses is the greatest common divisor of the numbers in that line.
• Rows and columns without a clue have no constraint of this kind.

Rows are r1..r6 from top to bottom and columns are c1..c6 from left to right. Clues above the grid are read downward, clues below it are read upward, clues on the left are read rightward and clues on the right are read leftward.

In words: r1 from the left has product 396. r3 from the right has product 456. r5 from the right has product 9. r6 from the right has GCD 5. c2 from the bottom has GCD 5. c3 from the top has product 2220. c6 from the top has product 2010.

Question. In the completed grid, the empty cells form several connected groups. What is the product of the areas of these groups?

Answer format: a single positive integer.