How many ordered pairs of positive integers $(m, n)$ satisfy the system:

\[
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\
\text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
\]

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

Let's think step by step and output the final answer within \boxed{}.
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