A point object of mass \(m\) is connected to a cylinder of radius \(R\) via a massless rope. At time \(t = 0\), the object is moving with an initial velocity \(v_0\) perpendicular to the rope. The rope has a length \(L_0\) and a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds \(T_{max}\). Express your answers in terms of \(T_{max}\), \(m\), \(L_0\), \(R\), and \(v_0\). The original answer is in the form \(\frac{k m^2 v_0^3}{n T_{max}}\). Please calculate the value of \(k + m + n\).

Let's think step by step and output the final answer within \boxed{}.
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