In triangle ABC, a circle with center O is inscribed and is tangent to sides AC, AB, and BC at points L, M, and N, respectively. Let D be a point on the arc MN, and from D, draw the tangent that intersects AB at point E, BC at point F, and the extension of AC at point G. Given that the angle COG measures 21 degrees and the angle CBG measures 24 degrees, calculate the measure of angle BGE.
Amidst points and arcs,
Inscribed circle's dance begins,
Angles weave their tale.
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