Let $\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).$ Then $(\mathcal{S} + 1)^3 = \frac{m}{n}$ with $m$ and $n$ coprime positive integers. Find $10m + n$.

Let's think step by step and output the final answer within \boxed{}.
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