Babylonian spiral

A Babylonian spiral is constructed by starting with a zero vector at the origin of a Cartesian grid and progressively concatenating the next longest vector with integer components. So the $i$th vector has integer components $(x_i, y_i)$ satisfying $x_i^2 + y_i^2 = n_i^2 > n_{i-1}^2$ where $x_i$, $y_i$ and $n_i$ are non-negative integers. The direction of the new vector is chosen such that it minimizes the clockwise angular separation from the previous one.

Starting with the vectors $(x_0, y_0) = (0, 0)$ and $(x_1, y_1) = (0, 1)$ find the $x$ and $y$ coordinates of the Babylonian spiral after $N$ steps.

Input: Number of steps $N$.

Output: The $x$ and $y$ coordinates as lists.