Let $f$ be a twice differentiable function such that $f(x)=\int_0^x \tan (t-x) d t-\int_0^x f(t) \tan t d t, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then $f^{\prime \prime}\left(\frac{\pi}{6}\right)+12 f^{\prime}\left(-\frac{\pi}{6}\right)+f\left(\frac{\pi}{6}\right)$ is equal to $\_\_\_\_$ .