How many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C$ in degrees are positive integers and satisfy the equation:

$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$

for some positive integer $k$, where $kC$ does not exceed $360^{\circ}$?

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