Jigsaw Thermometer Sudoku with no given numbers

the rightmost column has a thermometer of length 5, meaning the only possible sequence is 1,2,3,4,5.
columns: 12345
rows: a: ....5
b: ....4
c: ....3
d: ....2
e: ....1
because e5 is 1, the row e cannot have 1. however, the region still needs a 1, so that goes at the part that sticks up.
the same logic applies to the topmost region.
....5
5...4
....3
...12
....1
because the second-to-bottom region (largely consisting of row d) has only 3 tiles, those three must be 3, 4, and 5. so, the part that sticks out must be 1 and 2.
because of d4, we know there cannot be another 1 in column 4, so the 2 in the second-to-bottom region must be at c3.
the same logic applies to the second-to-top region (mostly row b).
....5
5...4
45123
...12
....1
because of the 5s in b1 and c2, the 5 in row d can only go at d3.
because of the 4 at c1, the 4 in row d can only go at d2.
so, the 3 must go at d1.
the same logic can be applied to row b.
....5
51234
45123
34512
....1
now, there is only one place each of the tiles can go; the rest of the options are blocked by the other numbers in the columns.
12345
51234
45123
34512
23451