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

\[ LCM(a,b,n) - (a+b+n) \]

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