Perfect triangular numbers

If we pick any even perfect number less than \(9^{9^{9^{9^{9^9}}}}\) at random, what is the probability that the perfect number chosen is also a triangular number?

Express the probability as a rational number \(\dfrac ab\), where \(a\) and \(b\) are coprime positive integers. And submit \(\mu\left(a+b\right)\cdot\left(ab\right)^2 \) as your answer.

Notation: \(\mu\) denotes the Möbius function.

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