Tree Properties at Successors of Singulars of Many Cofinalities

From many supercompact cardinals, we show that it is consistent for the tree property to hold at many small successors of singular cardinals, each with a different cofinality. In particular, we construct a model in which the tree property holds at $\aleph_{\omega+\omega+1}$ and at $\aleph_{\omega_n+1}$ for all $0<n<\omega$. We show that this can be done for the strong tree property as well, and extend the technique to large uncountable sequences of desired cofinalities.