Factorizations of simple groups of order 168 and 360

A finite group $G$ is called $k$-factorizable if for any factorization $\mathrm{\mid }G\mathrm{\mid }=a_1\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $\mathrm{\mid }{A}_i\mathrm{\mid }=a_i$ such that $G=A_1\cdots A_k$. We say that $G$ is $\mathit{\text{multifold-factorizable}}$ if $G$ is $k$-factorizable for any possible integer $k\ge 2$. We prove that simple groups of orders 168 and 360 are multifold-factorizable and formulate two conjectures that the symmetric group $S_n$ for any $n$ and the alternative group $A_n$ for $n\ge 6$ are multifold-factorizable.