Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 4abc.$ Find the maximum value of $$\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}.$$
First, I tried using $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}=\dfrac{4}{\sqrt{bc}}+\dfrac{6}{\sqrt{ac}}+\dfrac{12}{\sqrt{ab}}$. Then, I tried to apply some inequalities, but it didn't work.
I also tried just substituting $abc=\dfrac{a+b+c}{4}$ into the equation, but this also didn't do much. Could someone give me some guidance on how to proceed?
Thanks in advance!!!!