Twinned

In rectangle $ABCD, AD = 1, E$ is the midpoint of $BC$ , and $F$ is the midpoint of $AB$ . The incircles of $\triangle GED$ and $\triangle ECD$ are congruent. What is the value of $AB$ ?

If $AB$ is a root of $f(x) = ax^3 + bx^2 + cx + d$ , where $\gcd (a,b,c,d) = 1, a > 0$ . Submit $f(1)$ .