JEE Advanced 2024 Physics question on Rotational Motion

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse $P$ is imparted to the rod at a distance $x = L/n$ from the mid-point of the rod (see figure), then the rod string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of $n$ is _____.