HMMT 2025 A3

Given that $x$ , $y$ , and $z$ are positive real numbers such that

$$x^{\log_{2}(yz)} = 2^{8} \cdot 3^{4}, y^{\log_{2}(zx)} = 2^{9} \cdot 3^{6}, \text{and} z^{\log_{2}(xy)} = 2^{5} \cdot 3^{10},$$

compute the smallest possible value of $xyz$ .