#number theory inequality

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quick herald
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Let $n \in Z_{+}$ which has at least two different prime factors bigger than $3$. Let $m\geq 1$ be the smallest number such that $gcd(m(m+2), n) = 1$. Let $p$ and $q$ be two largest prime factors of $n$. Prove that $mpq \leq 5n$

fickle plazaBOT
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snaky_man

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shrewd lionBOT
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quick herald
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Any help would be greatly appreciated 🙂
I should probably mention that I played around with lcms but didnt get anything useful, other than that I just tried random approaches with no success...