Let \(y=y(x)\) be the solution of the differential equation \(x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\fr…

Let \(y=y(x)\) be the solution of the differential equation \(x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}\) and let \(\alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2+y^2-2 p x+2 p y+\alpha+2=0\) represents a circle of radius \(r \leq 6\), is \(\_\_\_\_\) .

[JEE Main 2026, 5 Apr (Shift 1)]