Provided you have no Verizon STB's in the house -- only TiVO -- you can do this. I have a Roamio, but it's the same principle.
You connect the coax to the Bolt, connect the 3rd party router to the ONT ethernet (and get Verizon to change the handoff from MoCA to Ethernet for Internet) and then either:
1. Install a MoCA bridge as the previous poster indicated such as the ECB2500 between a local LAN port on the router and the Coax cable plant (your Bolt needs a path to the internet for guide data).
2. Connect the Bolt via it's ethernet port to the local LAN and configure it for Ethernet vs MoCA in the setup.
You also remove and return the ActionTec router -- it's not needed.
Verizon does not support this setup (i.e. won't troubleshoot it since you don't have their router in the picture), but it works just fine.
Also, I would install a lo-pass line filter on the coax at the ONT if you use option #1 above to keep any MoCA signal from leaking out toward the ONT (or Verizon MoCA from coming in) just to be safe and insure no frequencies conflict.