Kinematics: JEE Main 2024 Physics question

A particle is moving in a circle of radius $50 \mathrm{~cm}$ in such a way that at any instant the normal and tangential components of it's acceleration are equal. If its speed at $\mathrm{t}=0$ is $4 \mathrm{~m} / \mathrm{s}$, the time taken to complete the first revolution will be $\frac{1}{\alpha}\left[1-e^{-2 \pi}\right] \mathrm{s}$, where $\alpha=$ _________.