# 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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