$(1+x)^2+(1+x)^3+(1+x)^4+\cdots+(1+x)^{49} + (1+mx)^{50}$

Let $m$ be the minimum positive integer such that the coefficient of $x^2$ in the expansion above is $(3n+1) \dbinom{51}{3}$, where $n$ is an integer.

Find $n$.

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