Missed this good slam. Over 3♠, we simply bid 4♣ – 5♣. Both could have done more.
Best is for East to make a “self-splinter” of 4♠, showing extras with a stiff spade.
Failing that, West could cue 4♦ over 4♣. This can’t be a suit. Hands with long diamonds would have bid 3♦.