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We propose a near-optimal relaying strategy for cooperative broadcast channels (CBC), CBC-SSAF, based on the class of sequential slotted amplify and forward (SSAF) strategies. Our strategy allows each destination to act in turn as a relay and forward its previously received signal to other destinations. While CBC-SSAF is not a full multiplexing gain strategy, the loss is negligible when the number of destinations is large. Moreover, CBC-SSAF allows each destination to be protected by the maximum number of extra paths in order to achieve the near-optimal diversity gain in the high multiplexing gain regime. A diversity and multiplexing tradeoff (DMT) lower bound for CBC-SSAF is derived which suggests that our proposed relaying strategy approaches the multiple-input multiple-output (MIMO) DMT upper bound and is therefore asymptotically optimal.