For certain real numbers $a$, $b$, and $c$, the polynomial

$g(x)=x^3+a{x}^2+2x+12$

has three distinct roots, and each root of $g(x)$ is also a root of the polynomial

$f(x)=x^4+x^3+b{x}^2+100x+c$.

What is $f(1)$?