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Are we bad for missing the slam, or is this just one of those slams that's tough to find with our current system?

Playing SAYC, 4SF, south dealer, opponents silent.

North:
S KQ43
H KT6
D KQ86
C K7

South:
S -
H A95
D A9542
C AQT95

1D 1S
2C 2H(1)
3C 3D
3H(2) 4D(3)
5D

1 - fourth suit forcing
2 - control showing bid
3 - denying first round control in spades or clubs.

Edit - corrected bidding.

V

You have a typo above -- North responded 1S, not 1H.

I have no idea how we get to 6D. Does my bidding 5C instead of 5D help at all?

I gave these hands to my husband and father-in-law, and here was their bidding:

1D 1S
2C 2H*
3C 3D
3H* 3NT
4D* 4NT
6D* 7D

2H was 4th suit forcing
3H was showing values in hearts
3NT was showing spade stoppers, and an option to play there
4D showed a stronger hand, a desire to continue bidding
4NT blackwood
6D was a special response -- it showed 3 or 0 keycards (in this case, 3), and a useful void (obviously spades, based on the bididng)
7D -- your hand correctly reasoned that whatever you needed, partner had it.

According to my husband, 7D was apparently the contract that we would like to have found on this hand.