The complete graph of \(y=f(x)\) which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is 1). Let \(a\) and \(b\) be the largest negative integer and the smallest positive integer, respectively, such that the functions \(g(x)=f(x)+ax\) and \(h(x)=f(x)+bx\) are invertible. What is \(a^2+b^2?\)
