A conductor whose resistance is constant \(R= 5 \ \Omega\) due to some reason is connected to a \(5 \ V\) battery. The heat lost \(Q\) to the surrounding due to radiations is dependent on both temperature of conductor \(T\) at any instant and time elapsed \(t\) as \(Q= a(T - T_0) +bt^2\), where \(T_0\) is temperature of conductor at time \(t=0\), \(a= 10 \ J/K\) and \(b= 4 \ J/s^2\). The conductor has a constant heat capacity.

Find the time after which the temperature of conductor becomes \(T_0\) again. If \(t= \dfrac XY\), where \(X\) and \(Y\) are coprime integers, find \(X+Y\).

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