Exponential lifetime of components in a system: probability system survives past time t

I am studying the following problem from an exam. A system consists of two subsystems connected in parallel. The first subsystem is just one component c1​. The second subsystem is made of three components c2,c3,c4 connected in series. The lifetimes of c1,…,c4​ are independent exponential random variables with rates λ1=0.11,λ2=0.03,λ3=0.01,λ4=0.06 (times in minutes). Compute the probability that the whole system is still working 9 minutes after switching on My reasoning was the following. Let Xi​ be the lifetime of component ci​, so P(Xi​>t)=e^−λi​t For the series subsystem c2,c3,c4 the lifetime is Y=min(X2​,X3​,X4​), so P(Y>t)=e^−(λ2​+λ3​+λ4​)t Since the first subsystem c1​ and the series subsystem are in parallel, the lifetime of the whole system is X=max(X1​,Y). Thus P(X>t)=1−P(X1​≤t,Y≤t)=1−(1−e−λ1​t)(1−e−(λ2​+λ3​+λ4​)t). So for t=9 $P(X>9)=e^{−9λ1}​+e^{−9(λ2​+λ3​+λ4​)}−e^{−9(λ1​+λ2​+λ3​+λ4​)}$ . With the given values I get P(X>9)≈0.631. However, when I discussed the exam with the professor, he said that I was mixing things, and that putting two exponential lifetimes in parallel does not give an exponential lifetime. I understand that the maximum of independent exponentials is not exponential, but I did not assume it was: I used the complement formula above. So my questions are: Is the value i get correct? If yes, why would the professor say that I was treating the parallel system as exponential? If not, where exactly is the mistake?