We have

$$\frac{(w - w_{1})}{(w_{1} - w_{2})} * \frac{(w_{2} - w_{3})}{(w_{3} - w)} = \frac{(z - z_{1})}{(z_{1} - z_{2})} * \frac{(z_{2} - z_{3})}{(z_{3} - z)}$$

$$\frac{(w + i)}{(-i + 1)} * \frac{(-1 - i)}{(i - w)} = \frac{(z + 1)}{(-1 -1)}* \bigg[\frac{\frac{z_{2}}{z_{3}} - 1}{1 - \frac{z}{z_{3}}}\bigg]$$

In order to adjust infinity

$$\therefore \bigg[\frac{w + i}{1 - w}\bigg]\bigg[\frac{-1 - i}{-i + 1}\bigg] = \frac{z + 1}{-2}(-1)$$

Solving by multiply it with complex conjugate or directly in calculator we get

$$\therefore \bigg(\frac{w + i}{i - w}\bigg)(-i) = \frac{z + 1}{2} \\ \therefore \bigg(\frac{w + i}{w - i}\bigg) = \frac{z + 1}{2i}$$

$\therefore$ By componendo and dividend we get

$$\therefore \frac{w + i + w - i}{w + i - w + i} = \frac{z + 1 + 2i}{z + 1 - 2i} \\ \therefore \frac{2w}{2i} = \frac{z + 1 + 2i}{z + 1 - 2i} \\ \therefore w = \frac{zi + i -2}{z + 1 - 2i}$$