If $2x+4(x-3)<-2-x<2x-1$, then find the value of $x$ from the given integer options.

The given combined inequality consists of two parts:
$$\text{Part 1: }2x+4(x-3)<-2-x$$
$$\text{Part 2: }-2-x<2x-1$$
Let's solve Part 1:
$$2x+4x-12<-2-x$$
$$6x-12<-2-x$$
$$7x<10⟹x<\frac{10}{7}\approx 1.43$$
Let's solve Part 2:
$$-2-x<2x-1$$
$$-2+1<3x$$
$$-1<3x⟹x>-\frac{1}{3}\approx -0.33$$
Combining both parts:
$$-\frac{1}{3}<x<\frac{10}{7}$$
Among the given options, only $x=1$ satisfies this inequality.