Why must the phase shift be divided by $4$ in order to properly graph the equation $f(x)=\sin(4x+\pi/3)$?

Question

Of course this is just an example of the intuition that I am having problems with. I understand that the period will be $2\pi/4$. This part is fine, but why must the phase shift $\pi/3$ also be divided by $4$. The hierarchy of operations is multiplication before addition. Therefore the "squeezing" of the period has already occurred, the period being $2\pi/4$. Then according to hierarchy of operations there is a phase shift of $\pi/3$, which means a shift to the left in this case, the opposite of the sign of $\pi/3$. What rule demands that we must first divide the phase shift by $4$, the same as we do to $2\pi$? This is very puzzling. If the expression read $f(x)=\sin\left(4\left(x+\frac{\pi/3}{4}\right)\right)$ then I would understand because by the hierarchy of operations. The expressions do not seem to me to have the same results on the sine function even though they are algebraically the same. My math book says to graph the function we must divide the phase shift by $4$ first after "squeezing" the period but I fail to see why.