Dividing Polynomials with 2 or More Unknowns

Alina is playing a game. She starts with a polynomial

$$x^4+a_1x^3y+a_2x^2y^2+a_3x^3y + y^4.$$

She flips a fair coin $3$ times. On the $n$-th flip, if she flips heads then $a_n = 1$ and if she flips tails then $a_n = 0$. For example, if she flips Head-Head-Tail then $a_1 = 1, a_2 = 1, a_3 = 0$. Alina wins if after $3$ flips, her polynomial is divisible by $x+y$. What is the probability that Alina wins?