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Re Breaking up GURPS turns
Actually, I am experimenting with discretely separate mental and physical speeds, though not for GURPS. I will certainly report here when I have something interesting!
As for declaring actions in reverse order: that's what was used for initiative in Star Frontiers. Initiative losers first declared what they were doing; winners next made their decisions based on that knowledge, and then took action first.
It's an interesting system that was fun (and possibly one of the few innovative things about SF combat). On the other hand, it's a little hard to explain in RL terms. For the Initiative "winner" to know what the "loser" is doing, the loser must begin the action while the winner waits and watches, right? So it all works like the GURPS Wait action – though in both cases, it's hard to explain why the guy who waited always acts before the guy who didn't wait.
Nits have been picked. That aside, the system you outline is pretty interesting, and doesn't bring any exotic new concepts into play. Nice. My biggest reservation, on first glance, would be an issue I briefly mentioned: directly tying DX and/or skill to number of actions per turn. In your system, a DX 20 guy would get twice the actions as DX 10, as well as the huge +10 on combat skills. Granted, DX 20 should be awesome-and-then-some, but that is adding a big bonus to DX that should be balanced by extra cost of DX.
Or maybe you're already assuming that standard 4e TH penalties apply for Rapid Strikes, which would place an appropriate penalty on Mr DX 20 as tried to take advantage of his amazing speed potential.
In any case, it'd be fun to put the system to the battleboard test!