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As each $P_A$ is a product of the elements of {$2,4,\dots,2^t$}, each element repeated at most $l$ times, we have for any $\lambda>0$, $$\sum_{A \in \mathcal{P}} P_A^\lambda\leq(2+4+\cdots+2^t)^{l\lambda}<2^{(t+1)l\lambda}<(4m^{1/r})^{l\lambda}.$$ Applying this for $\lambda:=r/l$ yields $$\sum_{A \in \mathcal{P}} P_A^{r/l} < 4^r m.$$