Is it true that $ a_k:= \sum_{n=1}^{k} \text{sgn}(\sin(n)) $ only takes values from $ \{ -2, -1, 0, 1, 2, 3 \},\ $ and if so, how do we prove it?

Question

Here, $\ \text{sgn}\ $ means the sign a.k.a signum function, and the angle units are in radians.

Based on experimentation (the Python code at the bottom), it seems that each member of the sequence $\ (a_k)_k\ $ defined by:

$$ a_k:= \sum_{n=1}^{k} \text{sgn}(\sin(n)) $$

only takes values from the set $ \{ -2, -1, 0, 1, 2, 3 \}. $

Is this true? And how can this be proven?

I know that $\ \displaystyle\sum_{n\in\mathbb{N} } \sin n\ $ is bounded, but I don't see how this helps. I also don't know about any of the equidistribution properties of the sequence $\ (\sin n)_n\ $ but I don't think this matters, as I doubt this would be useful because those things are asymptotic properties, not absolute properties.

But maybe I a overlooking something simple to answer my original question: however, I don't see it (obviously...). Here is my code:

import math
sineList=[]
for i in range(1000):
    if math.sin(i)>0:
        sineList.append(1)
    elif math.sin(i)==0:
        sineList.append(0)
    else:
        sineList.append(-1)
    print(sum(sineList))