in order to get FiOS TV you MUST use one of the ActionTec routers, it has to do with with MoCA as Steve indicated. you have however the option to lease or buy your own
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Not true. You only need the Actiontec if you get Verizon's set top boxes. If you go with a TiVo, you can put the Actiontec in the closet, and just get configured with Ethernet connectivity from the ONT.
To the OP, what steve said is correct. The Actiontec doesn't split the TV signal. It's just another end-point on your network, and it handles the IP feeds for your TV service. Those IP feeds would include things like guide data, widgets, and on-demand programming.
To split the TV signal, just a standard splitter will work fine. Although you will want one that can handle a 1GHz signal (the Actiontec handles VOD and multiroom feeds at ~900MHz, and many older splitters top out below that). You also don't want/need any amplifiers in your set up. The ONT runs very hot, and amplifying the signal is not only unnecessary but will cause break-ups. In fact, in my original set-up the installer had to put in an attenuator on one of my STBs.
Also, if you do get one of Verizon's STBs or DVRs, you will need the Actiontec in your set-up, but I know lots of folks that bridge other routers into the mix. Oh, and if you go for Verizon's media server, you will need the latest and greatest Actiontec (rev. I). It won't work without it.