Suppose $x+ \frac {1}{x}= \frac{4 \sqrt {3}}{3}$. Then $x^{10}+ \frac {1}{x^{10}}=a+ \frac {b}{c}$, where $a$, $b$ and $c$ are positive integers, $\frac{b}{c}$ is less than $1$, and $b$ and $c$ are coprime. Find $a+b+c$.

This problem is from the UP Math Wizard 2010.

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