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 ______.
