cool variables as hot constants !

Algebra Level 4

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

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


You can try my other Sequences And Series problems by clicking here : Part II and here : Part I.
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