KöMaL 2026 B5523

Find the largest possible number of distinct integer roots of polynomial $p(x)=a_{12}x^{12}+a_{11}x^{11}+\ldots+a_2x^2+a_1x+a_0$ with integer coefficients if $a_0+a_2+a_4+a_6+a_8+a_{10}+a_{12}=0$ and $a_1+a_3+a_5+a_7+a_9+a_{11}=12$.