Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a > b > c > d$ are positive integers satisfying:

$$|\zeta^a + \zeta^b + \zeta^c + \zeta^d| = \sqrt{3}.$$  

Compute the smallest possible value of $1000a + 100b + 10c + d$.

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