Two identical components having lifetimes $A$ and $B$ are connected in parallel in a system

.Suppose the distributions of $A$ and $B$ independently follow exponential with mean $\frac 1a, a>0$. But whenever one component fails the lifetime distribution of another changes to exponential with mean $\frac 1c ,c>0$ .Let $T$ denote the overall lifetime of the system

Find $P(T\ge t),\:t>0$

I am understanding that the total time will be $\min (A,B)+ \exp(c)$ but not getting how to write integral and also the limits of integration please help!!