Prime Triangle

Geometry Level 3

The figure above shows an isosceles triangle \(ABC\) with \(BA = AC \) and \(\angle BAC = 120^\circ\).

Let \(D\) be a point on \(AC\) such that \(AD > DC\). Given that the lengths of \(BA\), \(AD\), \(DC\), and \(BD\) are all distinct prime numbers, what is the length of \(BD\)?

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