Same Snapshot, Turned Camera

Two cameras photograph the same arrangement of arrows, each arrow stored as a complex number. One camera may be mounted turned by some angle \varphi, which multiplies every number by the same unit factor c = e^{i\varphi} (a complex number with |c| = 1). Turning keeps every length: a camera that zooms, which would multiply by a factor of another length, does not count.

Write same_up_to_unit(a, b) that returns True when there is one complex number c with abs(c) == 1 such that b[k] == c * a[k] for every index k (up to rounding), and False otherwise. In particular:

• lists of different lengths are never the same;
• two lists of zeros of the same length (and two empty lists) are the same, with c = 1;
• a zero must face a zero: [1, 0] and [1, 1] are not the same.

Later in this path, two quantum states that differ only like this will count as the same state.

The tests use a helper, turned(xs, phi), that returns [cmath.exp(1j * phi) * x for x in xs]. You can call it with Run too.

Examples

Input:  a = [1, 1j, 0], b = [1j, -1, 0]
Output: True
Explanation: c = 1j turns 1 into 1j and 1j into -1, and keeps 0 at 0.

Input:  a = [1, 1j], b = [2j, -2]
Output: False
Explanation: b = 2j * a, but |2j| = 2: that camera zoomed.

Input:  a = [1, 1], b = [1, -1]
Output: False
Explanation: the first arrow needs c = 1 and the second c = -1; one factor must fit all.

Constraints

• 0 <= len(a), len(b) <= 1000, and every entry has modulus at most 100
• the tests are never borderline: when the answer is True, b[k] and c * a[k] agree to within 1e-9; when it is False, every unit factor c misses some entry by more than 0.001. A tolerance such as 1e-6 is safe.

Goals

• Decide whether two lists differ by one common factor of modulus 1
• Find the candidate factor from a safe entry instead of dividing by 0
• Compare floating-point complex numbers with a tolerance