Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. 

You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.

Let's think step by step and output the final answer within \boxed{}.
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