\(ABCDEFGH\) is a regular octagon. Express in terms of a single vector the sum of the vectors

\(\mathbf{AC},\mathbf{AH},\mathbf{HG}\),

\(\mathbf{AB},\mathbf{AC},\mathbf{AF},\mathbf{AH},\mathbf{HG}\).

Given that \(\big \vert \mathbf{AB} \big \vert = 2,\) calculate \(\big \vert \mathbf{AF} \big \vert\).