Comet and Earth Encounter

Find the maximum time a comet following a parabolic trajectory around the Sun can spend within the orbit of the Earth. Assume that the Earth’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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