Cube in a cone

Geometry Level pending

A cone has a circular base of radius 120 centimeters and center \(O\), and a vertex \(V\) that is \(240\sqrt{2} \) centimeters directly above \(O\). A cube is placed such that it has 4 vertices on the circular base, and 4 vertices on the lateral side of the cone. The side length of the cube can be expressed as \( a \sqrt{b} \), where \(a\) and \(b\) are positive integers, and \(b\) is not divisible by the square of any prime. What is \(a+b\)?

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