A sector of a circle of radius \(10\) and subtended angle \(60°\) has a circle inscribed in it - its diameter can be written as \(\frac { a }{ b } \), where \(a\) and \(b\) are co-prime integers.
###### Part 1: Circle-ception.

Find the value of\[\left\lceil \frac { { a }^{ 2 }b^{ 2 } }{ a+b } \right\rceil \]

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