(1) Artificial, shows a good hand of 16+ points. (Think Gazilli)
(2) 6+♠, good suit
I'm trying not to bring system into this, but of course it has some bearing. You can think of an equivalent problem in your own systems. Here partner could have bid 2♣ over your 1NT bid to find out more, but has chosen to emphasize his own suit. For what it's worth, your 1♦ opening showed 4+ and since you did not splinter or make a direct raise, your artificial 1NT shows that you have 5+♦. Finally 3♠ "sets" spades. So here are your options:
3NT Non-serious slam try for ♠
4♣ Serious slam try for ♠, shows 1st/2nd round control
4♦ Serious slam try for ♠, shows 2/3 top ♦ honors
4♠ To play
4NT RKCB
5♣ EKCB
Can't think of any other viable alternatives. What do you think is the best plan of attack from here? What are your continuations over partner's likely responses?

Help

1NT(1) - 3♠(2)
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