Find the least possible value of $\dfrac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.
Source: Argentina Cono Sur TST 2024 P4
Tags: algebra
Find the least possible value of $\dfrac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.