Product construction with two levels proposed in  is a lattice construction which can be thought of as Construction A with codes that can be represented as the Cartesian product of two linear codes. This paper first generalizes the product construction to arbitrary number of levels. More importantly, the existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is proved. This family of lattices is then used to generate a sequence of nested lattice codes based on the recent construction of Ordentlich and Erez. This allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward with multistage decoding, which is termed multistage compute-and-forward.