Differential Equations: JEE Main 2026 Mathematics question

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(t))^2+(f'(t))^2\right)dt$. Then the mean of $f(\log_e(1)), f(\log_e(2)), \ldots, f(\log_e(625))$ is equal to ______.