Precariously Perched Laptop

A student is preparing feverishly for final exams, and he doesn't realize that his laptop has been creeping towards the edge of his desk and is about to fall off.

What fraction of the base of the laptop can hang off of the edge of the desk before the entire thing falls off?

Details and Assumptions:
\hspace{0.5cm}\bullet The base of the laptop and the top (screen) portion both have the same length LL.
\hspace{0.5cm}\bullet Both the base and the screen have uniform mass density as a function of length.
\hspace{0.5cm}\bullet The base of the laptop is twice as massive as the top (screen) portion.
\hspace{0.5cm}\bullet The screen makes an angle of 120 degrees with respect to the base.
\hspace{0.5cm}\bullet Neglect the thickness of the base and of the screen.
\hspace{0.5cm}\bullet The 120-degree angle mentioned above is the only non-right angle in this problem.
\hspace{0.5cm}\bullet Treat the laptop, as a whole, as a rigid body.
\hspace{0.5cm}\bullet Give your answer as a decimal number between 0 and 1.

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