Asymmetric Penalties in an Exponential Race

Two independent alarms go off at $\operatorname{Exp}(4)$ and $\operatorname{Exp}(6)$ times respectively. If alarm 1 fires first you pay $\$3$; if alarm 2 fires first you pay $\$5$. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an additional $\$1$ when it fires. Find the expected total payment.