555 Timer with controllable reset
You can do that, with a bit of effort.

However, you're really crossing into regions where it's much easier to simply learn how to use an arbitrary low-cost microcontroller (e.g. an Arduino-compatible) and program exactly what you want. Much more flexible, much less potential for analog headache!



To achieve that, you could catch the falling edge of the other circuit and use it to discharge C1. 

Not having tried that, but: 

I think it would be possible to achieve using a capacitor to couple in the falling edge to the base of an NPN transistor, which you use as inverter, to another drive another NPN's base whose collector-emitter is in parallel with C1.