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The Actiontec device is a router, not a modem. This is an important distinction, because multiple routers on one network require care to make sure they play nicely with each other. This translates into making sure they both aren't trying to manage the network; if the are, things break. Which sounds like your problem..
When you plug your new router into your existing network, make sure you use the "WAN" port not a "LAN" port. A "WAN" port provides a router with the connection to the internet. "LAN" provides local devices with access to the internet. If you connect two routers via their LAN ports, th will both attempt to provide a connection to any other devices. This will take the network down.
When properly connected (LAN->WAN), a second router can create it's own network. Communication between devices on the different routers will not be possible. To avoid that, you need to configure the 2nd router to act as an access point, not a router. In particular, it needs it's DHCP server disabled and maybe some more changes to other settings.
For way more details, check out the FiOS FAQs on broadbandreports.com:
http://www.dslreports.com/faq/verizonfios
You could also try Apple support or their forums.
Good Luck!