Find the maximum time a comet following a parabolic trajectory around the Sun can spend within the orbit of the Earth. Assume that the Earthâ€™s orbit is circular and in the same plane as that of the comet.

The answer is of the form \( \dfrac{aT}{b\pi} \), where \(T\) is the time period of Earth's revolution, and \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

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