I am Groot!

Algebra Level 2

A tree undergoes a sprouting growth.

Its longest branch is 32 units long, and the pair of branches connected to it are both $\frac{1}{3}$ the length of that branch. The same pattern continues with the length consistently shortening by $\frac{1}{3}$ in each step. Adjacent branches meet at $120^\circ.$

If this tree sprouts an infinite number of branches, what is its total height?

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