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$\begin{cases} a+b+c+d=12 \\ abcd=27+ab+ac+ad+bc+bd+cd \end{cases}$

If the above equations hold true simultaneously for some positive real numbers $a,b,c$ and $d$, then find the value of $\left( \dfrac{a^4.b^5.c^6}{2.d^{14}} \right)$

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