Compute the number of ordered pairs of integers \((a, b)\), with \(2 \le a, b \le 2021\), that satisfy the equation
\[
a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.
\]

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