$$^{238}_{92}U$$ is known to undergo radioactive decay to form $$^{206}_{82}Pb$$ by emitting alpha and beta particles. A rock initially contained $$68 \times 10^{−6}$$ g of $$^{238}_{92}U$$ If the number of alpha particles that it would emit during its radioactive decay of $$^{238}_{92}U$$ to $$^{206}_{82}Pb$$ in three half-lives is Z \times 10^{18}, then what is the value of Z?