Triangle \(ABC\) is an acute triangle. Let \(E\) be the foot of the perpendicular from \(B\) to \(AC\), let \(F\) be the foot of the perpendicular from \( C\) to \(AB\). If \( 2[AEF] = [ABC] \), what is the measure (in degrees) of \(\angle BAC \)?

**Details and assumptions**

\([PQRS]\) denotes the area of figure \(PQRS\).

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