HMMT 2024 A8

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

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

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