CD perpendicular to AB

Geometry Level 4

Let \(ABC\) be a triangle right angled at \(C\). We are given that \(CD \perp AB\) and \(F \) is the center of the inscribed circles of the triangles \(ADC\) and \(BDC\). Parallel lines through \(E\) and \(F\) with \(CD\) meet \(AC\) and \(BC\) at points \(E'\) and \(F'\).

Denote \(n = \frac{CE'}{CF'} \) and that \(n \) is a rational number of the form \(\frac pq\) for coprime positive integers \(p\) and \(q\). Find the value of \(p^3 + q^3\).

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