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Let $X_{n}$ be an $n \\times n$ random matrix over $\\mathbb{Z}_{p}$ with independent entries that lie in any residue class modulo $p$ with probability at most $1 - \\epsilon$ for a fixed real number $0 < \\epsilon < 1$. We prove that as $n \\rightarrow \\infty$, the distribution of the cokernel $\\mathrm{cok}(P(X_{n}))$ of $P(X_{n})$ converges to the distribution given by a finite product of some explicit measures that resemble Cohen--Lenstra measures. 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Let $X_{n}$ be an $n \\times n$ random matrix over $\\mathbb{Z}_{p}$ with independent entries that lie in any residue class modulo $p$ with probability at most $1 - \\epsilon$ for a fixed real number $0 < \\epsilon < 1$. We prove that as $n \\rightarrow \\infty$, the distribution of the cokernel $\\mathrm{cok}(P(X_{n}))$ of $P(X_{n})$ converges to the distribution given by a finite product of some explicit measures that resemble Cohen--Lenstra measures. 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