Let \( f \) be a differentiable function defined on \( \left[0,\frac{\pi}{2}\right] \) such that \( f(x) > 0 \) and \( f(x) + \int_{0}^{x} f(t)\sqrt{1-(\log_{e} f(t))^{2}}\,dt = e, \ \forall x \in \left[0,\frac{\pi}{2}\right] \). Then \( \left(6 \log_{e} f\!\left(\frac{\pi}{6}\right)\right)^{2} \) is equal to ________.
