Find the sum of the roots of the exponential equation

Question

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum.

I'll simplify it first as:

$$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$

Let $u = 2^{111x}$ then:

$$\frac{u^3}{4} + 4u = 2u^2 + 1$$

$$u^3 - 8u^2 + 16u - 4 = 0$$

The solutions are $u_1, u_2, u_3$

$$u_1 + u_2 + u_3 = 8$$

But that isnt for the original.

Please, only small HINTS, no answers please, thanks!

Answer 1

HINT :

$$111x=\log_2(u)\Rightarrow x=\frac{\log_2(u)}{111}.$$

Answer 2

Hint: The product of the roots of the new cubic is $2^{111(x_1 + x_2 + x_3)}$, where $x_1, x_2, x_3$ are the roots.

This is 2005 AIME I #8.