Area in complex plane

Geometry Level 4

Let \(S = S_1 \cap S_2 \cap S_3\) where:

  • \(S_1 = \left\{ z \mid z \in \mathbb{C}, |z| < 4 \right\}\)
  • \(S_2 = \left\{ z \mid z \in \mathbb{C}, \Im \left( \frac{z - 1 + i \sqrt{3}}{1 - i \sqrt{3}} \right) > 0 \right\}\)
  • \(S_3 = \left\{ z \mid z \in \mathbb{C}, \Re(z) > 0 \right\}\)

If the area of \(S\) can be expressed as \(\dfrac{a}{b} \pi\), where \(a\) and \(b\) are positive integers that are relatively prime, find \(a+b\).

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