For certain real numbers $a$ , $b$ , and $c$ , the polynomial $$g(x) = x^3 + ax^2 + x + 10$$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $$f(x) = x^4 + x^3 + bx^2 + 100x + c.$$ What is $f(1)$ ?