ALIBABA 2021 Problem 18

Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$ -module of rank $2m+1$ , equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$ -bilinear form $$( ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.$$ Here ``perfect'' means that the induced map $$\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), x \mapsto (x,\cdot)$$ is an isomorphism. Find the cardinality of the set $$\{x \in \Lambda: (x,x)=0\},$$ expressed in terms of $p,m,N$ .