Prelims
a) 6(3) cage at R1C1 = {123}
b) 21(3) cage at R1C7 = {489/579/678}, no 1,2,3
c) 22(3) cage at R2C2 = {589/679}
d) 9(3) cage at R3C3 = {126/135/234}, no 7,8,9
e) 11(3) cage at R4C2 = {128/137/146/236/245}, no 9
f) 10(3) cage at R6C6 = {127/136/145/235}, no 8,9
g) 20(3) cage at R7C7 = {389/479/569/578}, no 1,2
h) 20(3) cage at R8C1 = {389/479/569/578}, no 1,2
i) 20(3) cage at R8C4 = {389/479/569/578}, no 1,2
j) 10(3) cage at R8C6 = {127/136/145/235}, no 8,9
1a. Naked triple {123} in 6(3) cage at R1C1, locked for N1
1b. 22(3) cage at R2C2 = {589/679}, 9 locked for N1
1c. 9(3) cage at R3C3 = {126/135/234}
1d. R3C3 = {456} -> no 4,5,6 in R3C4 + R4C3
2a. 45 rule on R89 2 innies R8C27 = 11, no 1,9 in R8C2
2b. 45 rule on C89 2 innies R27C8 = 13 = {49/58/67}, no 3
2c. 45 rule on N3 3 innies R3C789 = 10 = {127/136/145/235}, no 8,9
2d. 45 rule on N3 1 outie R4C9 = 1 innies R3C7 + 6 -> R3C7 = {123}, R4C9 = {789}
2e. 45 rule on N7 3 innies R789C3 = 9 = {126/135/234}, no 7,8,9
2f. Killer triple 1,2,3 in R4C3 and R789C3, locked for C3
2g. Min R5C3 = 4 -> max R45C2 = 7, no 7,8 in R45C2
2h. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 4 -> R7C3 = {12345}, R9C4 = {56789}
2i. 45 rule on N124 2(1+1) outies R3C7 + R7C3 = 4 = [13/22/31], clean-up: no 8,9 in R9C4
2j. Max R7C3 = 3 -> min R6C23 = 14, no 1,2,3,4 in R6C2
2k. 15(3) cage at R2C6 = {159/168/249/258/267/348/357} (cannot be {456} because R3C7 only contains 1,2,3)
2l. R3C7 = {123} -> no 1,2,3 in R23C6
2m. 45 rule on R6789 2 innies R6C15 = 7 = {16/25/34}, no 7,8,9
2n. 45 rule on C6789 2 innies R15C6 = 13 = {49/58/67}, no 1,2,3
2o. 45 rule on N9 2 innies R7C9 + R9C7 = 9, no 1,9 in R7C9
2p. 45 rule on R12 4 outies R3C2567 = 24
2q. Max R3C7 = 3 -> min R3C256 = 21, no 1,2,3 in R3C5
2r. Hidden killer triple 1,2,3 in R3C4 and R3C789 for R3, R3C4 = {123} -> R3C789 must contain two of 1,2,3 = {127/136/235}, no 4 in R3C89
2s. 45 rule on C12 2 outies R25C3 = 1 innie R6C3 + 8, IOU no 8 in R2C5
[The first key step.]
3a. 45 rule on N4 2 outies R3C1 + R7C3 = 1 innie R4C3 + 6, IOU no 6 in R3C1
3b. R3C7 + R7C3 = [13/22/31] (step 2i)
3c. Consider combinations for 9(3) cage at R3C3 = {126/135/234}
9(3) cage = {126/234} => 2 placed in R3C4 or R4C3 => R3C7 + R7C3 = [13/31]
or 9(3) cage = {135} => R3C3 = 5 => R3C1 = {48} (R3C13 cannot be [75] which clashes with 22(3) cage at R2C2), R4C3 = {13} => R7C3 must be the reverse {13}
[Note that with R3C1 = {48} and the outie-innie difference of 6, R4C3 and R7C3 have to differ by +2 or -2 in this path]
-> R3C7 + R7C3 = {13}, no 2, clean-up: no 8 in R4C9 (step 2d), no 6 in R9C4 (step 2h)
3d. Naked pair {13} in R3C7 + R7C3, CPE no 3 in R7C7
3e. R3C789 (step 2r) = {136/235} (cannot be {127} = 1{27} because 16(3) cannot be {27}7), no 7, 3 locked for R3 and N4
[Taking things a bit further.]
3f. 45 rule on N5689 2(1+1) innies R4C9 + R9C4 = 14 = [77/95]
3g. Consider placements for R4C9
R4C9 = 7 => R3C89 = 9 = {36}
or R4C9 = 9 => R9C4 = 5 => R89C3 = 8 = {26}, locked for C3 => 9(3) cage at R3C3 = [423/513] => R3C789 = {136}
-> R3C789 = {136}, 1,6 locked for R3 and N3
[Fairly straightforward from here; no further clean-ups.]
3h. R3C4 = 2 -> R34C3 = 7 = [43], R7C3 = 1, R3C7 = 3, R3C89 = {16} = 7 -> R4C9 = 9 (cage sum), R9C4 = 5
3i. Naked pair {26}, locked for N7, 6 locked for C3
3j. R2C2 = 6 (hidden single in N1) -> R2C3 + R3C2 = 16 = {79}, 7 locked for N1
3k. R47C3 = [31] -> R3C1 = 8 -> R1C3 = 5
3l. 11(3) cage at R4C2 = {128} (only remaining combination, cannot be {146/245} because R5C3 only contains 7,8) -> R5C3 = 8, R45C3 = {12}, locked for C3 and N4 -> R1C2 = 3
3m. R7C3 = 1 -> R6C23 = 16 = {79}, locked for R6 and N4
3n. Naked triple {456} in R456C1, 4,5 locked for C1
3o. Naked pair {79} in R36C2, locked for C2
4a. R3C7 = 3 -> R23C6 = 12 = {57} (cannot be {48} because 4,8 only in R2C6), locked for N2 -> R3C5 = 9, R3C2 = 7 -> R23C6 = [75], R2C3 = 9
4b. R3C5 = 9 -> R2C45 = 4 = {13}, 1 locked for R2 -> R2C1 = 2
4c. 21(3) cage at R1C7 = {489} (only remaining combination, cannot be {579} because 7,9 only in R1C7) -> R1C7 = 9, R2C78 = {48}, locked for N3 -> R2C9 = 5
4d. 1 in N8 only in R89C6, locked for C6 and 10(3) cage at R8C6
4e. 10(3) cage at R8C6 = {127/136}, no 4
4f. 7 on {127} must be in R9C7 -> no 2 in R9C7
4g. R9C7 = {67} -> no 6 in R89C6
4h. R7C9 + R9C7 (step 2o) = 9 = [27/36]
4i. R7C9 = {23} -> R6C89 = 13,14 = {58/68}, 8 locked for R6 and N6
4j. R5C6 = 9 (hidden single in C6) -> R56C5 = 8 = {26/35}/[71], no 4, no 1 in R5C4
4k. 10(3) cage at R6C6 = {136} (only remaining combination, cannot be {145} because 1,5 in R6C7, cannot be {235} because R67C6 = {23} clashes with 10(3) cage at R8C6) -> R6C7 = 1, R67C6 = {36}, locked for C6
4l. Naked pair {12} in R89C6, 2 locked for C6, R9C7 = 7 -> R7C9 = 2
4m. R7C9 = 2 -> R6C89 = 14 = {68}, 6 locked for R6 and N6 -> R67C6 = [36], R6C145 = [542], R5C5 = 6 (cage sum)
4n. R4C6 = 8 -> R45C7 = 6 = {24}, locked for N6, 4 locked for C7 -> R2C7 = 8
4o. R78C7 = [56] -> R7C8 = 9 (cage sum)
4p. R6C4 = 4 -> R7C45 = 11 = {38}, locked for R7 and N8
and the rest is naked singles.