How many ordered pairs $(a, b)$ of positive integers satisfy the equation 
\[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\]
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?

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