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Rebidding with a fit after partner's 3 level invitational jump shift

We play 3-level jump shift responses to 1 level suit openers as natural, invitational, 6+ card suit.

Over these, we play all non-game rebids by opener as forcing (e.g. 1-3-3).

But I'm not quite sure how to show a hand with a good fit for responder with enough strength to have slam interest after the auction starts:

1-3.

When partner's jump shift is into a minor, the solution is that a raise to 4 of the minor is forcing with slam interest.

But obviously that does not work after 1-3.

If opener is strong enough to take control, he can just jump to 4 Kickback key-card ask (with long strong s and no fit, opener just rebids 3 forcing, then 4).

But what can opener do to show slam interest in s but not enough to launch into key-card ask?

Certainly jumping to 5 is possible, but that is a rather ugly call.

I'm wondering whether one or both of opener's 4m rebids (4 and/or 4) should be appropriated to show slam interest in s rather than being used as natural forcing bids to show a strong 2-suiter.

As an example, a hand I held today (matchpoints) was:  AQJT8-K72-AQ87-6, and our auction started 1-3.

On this particular example, I suppose I could have tried a jump to 5 which I hope partner would read as a splinter (for s ??). But instead I just made a cowardly raise to 4.  Partner's hand was: 4-AQ8654-J953-K3 (admittedly not exactly typical for 3).

6 was makeable via a ruffing finesse in s, but after T lead (yes it was a stiff), partner rose A, pulled trump in 2 rounds and just conceded a to make 5.  The deal was played 9 times:  no one bid slam, only 5 pairs even reached game, and none made more than 5.

Anyway, I'd be interested in your views as to whether 4 and/or 4 rebids by opener ought to be used to show interest in a slam.  If not, what alternatives would you suggest? 

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