Mystic cevians

Geometry Level 5

Let \(H\) be the orthocentre and \(O\) the circumcentre of a triangle \(ABC\) with the length of side \(BC\) equal to \(2014\). Cevians are constructed from vertices \(B\) and \(C\) so as to be tangent to \(\gamma\) , the circle passing through \(A\) and \(H\) and congruent to the circle centred at \(O\) touching \(BC\).

Let the cevians through \(B\) and \(C\) touch the circle \(\gamma\) at \(T\) and \(U\) respectively. If \(BT\) and \(CU\) are of integer lengths, what is their sum?

Aside: On the query of some, this is indeed an original problem of mine!

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