Professor Jablonski of the Kersenboom Institute likes to combine her expertise in geometry with a casual interest in fruit trees. One evening while reading the Wikipedia page on citrus taxonomy she stumbles across the following idea.

There are three pure types of citrus tree, the pomelo, the citron and the true mandarin. These are represented as corners of an equilateral triangle, with the pomelo at $a=(0,0)$, the citron at $b=(1,0)$ and the true mandarin at $c=(\frac{1}{2}, \frac{\sqrt{3}}{2})$. Other species of citrus trees are just combinations of the three pure types. For example, the Bergamot Orange is $40\%$ pomelo, $30\%$ citron and $30\%$ true mandarin, and so can be represented by the point $$(0.4\times a)+(0.3\times b)+(0.3\times c)=(0.45, \frac{3\sqrt{3}}{20}).$$

In year 1 Jablonski starts with a pomelo. She breeds her pomelo once with another pomelo, once with a citron and once with a true mandarin to create, at year 2, trees represented by points $\frac{a}{2}+\frac{a}{2}, \frac{a}{2}+\frac{b}{2}$ and $\frac{a}{2}+\frac{c}{2}$.

In year $n$ she breeds each of her trees once with a pomelo, once with a citron and once with a true mandarin. This means that each tree at point $t$ in year $n$ will have children in year $n+1$ at points $\frac{t}{2}+\frac{a}{2}, \frac{t}{2}+\frac{b}{2}$ and $\frac{t}{2}+\frac{c}{2}$.

Jablonski only considers two types of citrus to be different if their geometric representations are distance at least $2^{-10}$ apart. What is the maximum number of different citruses she can make by following her breeding plan?

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