A tower system

Algebra Level 4

$\huge x^{y^{x^{y^{x^{y^{.^{.}}}}}}} = 2 \quad,\quad y^{x^{y^{x^{y^{x^{.^{.}}}}}}} = \frac{3}{2}$

Let $x$ and $y$ be real numbers such that they satisfy both the equations above. If $xy$ can be expressed in the form $\sqrt[n]{\frac{a}{b}}$ where $gcd(a,b)=1$ and $n$ is the minimum positive integer satisfying the expression, determine