If necessary modify the choice of $\pi _ i$ such that for $i = 1, \ldots , d$ we have
\[ \text{Ann}_{R/(\gamma _1^ e, \ldots , \gamma _{i - 1}^ e)R}(\gamma _ i) = \text{Ann}_{R/(\gamma _1^ e, \ldots , \gamma _{i - 1}^ e)R}(\gamma _ i^2) \]
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