r}{N\choose m}{\ell-0 \choose m-0}}{\text{poly}(nrs){n+r \choose r}{N \choose m+rs}{\ell+n-r \choose m+rs}}$$  This bound holds for any choice of positive integers $\ell, m, r$, a constant $\beta$ such that $s = \beta\log n$ which satisfy the constraints in the hypothesis of Corollary \[cor:circuit-complexity-bound\] and Lemma \[lem:manyshifts