Area in complex plane

Geometry Level 4

Let \(S={ S }_{ 1 }\cap { S }_{ 2 }\cap { S }_{ 3 }\)

where

\[\large{\begin{cases} { S }_{ 1 }=z\in { C };\left| z \right| <4 \\ { S }_{ 2 }=z\in { C;Im\left( \frac { z-1+\sqrt { 3 } i }{ 1-i\sqrt { 3 } } \right) >0 } \\ { S }_{ 3 }=z\in { C };Re(z)>0 \end{cases}}\]

If the area of \(S\) can be expressed as \(\frac{a}{b}\pi\) where \(a,b\) are co prime natural numbers.

Find \(a+b\)

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