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 \( x=\frac{L}{n} \) from the midpoint of the rod (see figure), then the rod and string revolve together around the point \( O \), with the rod remaining aligned with the string. In such a case, the value of \( n \) is ________.
