calculus - Which part of the train stays for the longest time in the station?

A train with length $L$ is moving towards a train station of length $S$ with speed $v$.

The train starts to decelerate with acceleration $-a$ as soon as head reaches the station until completely stops. Right after the train completely stops, it starts to accelerate with acceleration $a$ until its tail leaves the station with speed $v$.

Which part of the train stays for the longest time in the station?

The correct answer is the mid-part of the train. If the head stays in the station for $t$ and the mid-part of the train stays for $\sqrt{2}t$. Could anyone give me some clue on how to get this?