Proportional square

Geometry Level 3

\( ABCD \) is a square. A point \( G \) is placed on the side \( BC \) so that \( \frac{BG}{GC} = \frac{1}{3} \) and the midpoint of the side \( CD \) is \( E \). Segments \( BE \) and \( AG \) have a common point \( X \).

If \( \frac{BX}{BE} = \frac{a}{b} \) where \( a \) and \( b \) are positive coprime integers, Find \( a + b \).

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