Analysis
This problem tests our understanding of the arithmetic rules for complex numbers, making it a fundamental question.

Step-by-Step Solution
Recall the sum of a geometric series formula:
$$S_n = \frac{a_1(1-r^n)}{1-r}$$
where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

In the given problem, $a_1=i$, $r=i$, and $n=12$.

Applying the formula, we have:
\begin{align*} z &= \frac{i(1-i^{12})}{1-i} \\ &= \frac{i(1-(i^4)^3)}{1-i} \\ &= \frac{i(1-1)}{1-i} \quad \text{(since i^4 = 1)} \\ &= \frac{i(0)}{1-i} \\ &= 0 \end{align*}

Therefore, the answer is $\boxed{0}$.