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 _______.

If the solution curve $y=f(x)$ of the differential equation

$$ \left(x^2-4\right) y^{\prime}-2 x y+2 x\left(4-x^2\right)^2=0, x>2, $$

passes through the point $(3,15)$, then the local maximum value of $f$ is $\_\_\_\_$

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) \mathrm{dt}$. Then the mean of $f\left(\log _{\mathrm{e}}(1)\right), f\left(\log _{\mathrm{e}}(2)\right), \ldots . ., f\left(\log _{\mathrm{e}}(625)\right)$ is equal to $\_\_\_\_$ .