It's not completely inscribed!

Geometry Level 4

Let \(ABC\) be an equilateral triangle. Let \(D\) be on \(BC\), \(E\) and \(F\) be on \(AB\) and \(G\) be outside \(ABC\) such that \(DG \mid \mid EF\) and \(\angle DEF = \angle GFE = 90^{\circ}\). Let \(AC\) intersect \(DG\) at \(J\) and intersect \(FG\) at \(K\). It is known \(CD=25\) and \(JK=16\). Find the length of \(EF\).

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