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