Let P(x, y) be a polynomial with real coefficients in the variables x, y that is not identically zero. Suppose that P(\lfloor 2 a\rfloor,\lfloor 3 a\rfloor)=0 for all real numbers a. If P has the minimum possible degree and the coefficient of the monomial y is 4, find the coefficient of x^{2} y^{2} in P. (The degree of a monomial x^{m} y^{n} is m+n. The degree of a polynomial P(x, y) is then the maximum degree of any of its monomials.)