Before revision:  We show that the cost of the strategy (3^k, ,3) with k\approx \log_3(1/p) is of order p\log(1/p), and differs from the optimal cost by a fraction of this value. To prove this result we bound the difference between the cost of this strategy with the minimal cost when pool sizes take real values. 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 show that the cost of the strategy (3^k, for p\in (0, 1-3^{-1/3 (3^k , , 3 ) or of the form (3 ^{k-1}4,3^{k-1} \dots,3 ), with a precise description for k.