Two moving points from the origin

Calculus Level 2

Two points \(P\) and \(Q\) start at the origin and move from the origin at the same time along the \(x\)-axis. At time \(t>0\), \(P\) and \(Q\) have velocities of \(\sin (\pi t)\) and \(2\sin (2\pi t)\), respectively. If \(P\) and \(Q\) first meet again after \(\frac{m}{n}\) seconds, where \(m\) and \(n\) are positive co-prime integers, what is the value of \(m+n\)?

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