Stones, Up and Down

From an elevated point \(A\), a stone is projected vertically upwards. When the stone reaches a distance \(h\) below \(A\), its velocity is double of what it was at a height \(h\) above \(A\). If the greatest height attained by the stone is \(\frac{ph}{q}\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

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