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