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Let \( ABC \) be a triangle with \( AB = 2 \), \( AC = 3 \), and \( BC = 4 \). The isogonal conjugate of a point \( P \), denoted \( P^{*} \), is the point obtained by intersecting the reflection of lines \( PA \), \( PB \), and \( PC \) across the angle bisectors of \( \angle A \), \( \angle B \), and \( \angle C \), respectively.

Given a point \( Q \), let \( \mathfrak{K}(Q) \) denote the unique cubic plane curve which passes through all points \( P \) such that line \( PP^{*} \) contains \( Q \). Consider:
- the M'Cay cubic \( \mathfrak{K}(O) \), where \( O \) is the circumcenter of \( \triangle ABC \),
- the Thomson cubic \( \mathfrak{K}(G) \), where \( G \) is the centroid of \( \triangle ABC \),
- the Napoleon-Feurerbach cubic \( \mathfrak{K}(N) \), where \( N \) is the nine-point center of \( \triangle ABC \),
- the Darboux cubic \( \mathfrak{K}(L) \), where \( L \) is the de Longchamps point (the reflection of the orthocenter across point \( O \) ),
- the Neuberg cubic \( \mathfrak{K}(X_{30}) \), where \( X_{30} \) is the point at infinity along line \( OG \),
- the nine-point circle of \( \triangle ABC \),
- the incircle of \( \triangle ABC \), and
- the circumcircle of \( \triangle ABC \).

Estimate \( N \), the number of points lying on at least two of these eight curves. An estimate of \( E \) earns \( \left\lfloor 20 \cdot 2^{-|N - E| / 6} \right\rfloor \) points.

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