Before revision:  We conjecture that the optimal strategy, depending on the value of p, is indeed of the form (3^k , , 3 ) or of the form (3 ^{k-1}4,3^{k-1}  \dots,3  ), with a precise description for k.
After revision:  We conjecture that the optimal strategy, depending on the value of p, is indeed of the form (3^k , , 3 ) or of the form (3 ^{k-1}4,3^{k-1} \dots,3 ^2,3 ), with a precise description for k.