A plane geometric figure of $n$ sides with the vertices $A_1,A_2,A_3,\dots, A_n$ ($A_i$ is adjacent to $A_{i+1}$ for every $i$ integer where $1\leq i\leq n-1$ and $A_n$ is adjacent to $A_1$) is called brazilian if: I - The segment $A_jA_{j+1}$ is equal to $(\sqrt{2})^{j-1}$, for every $j$ with $1\leq j\leq n-1$. II- The angles $\angle A_kA_{k+1}A_{k+2}=135^{\circ}$, for every $k$ with $1\leq k\leq n-2$. Note 1: The figure can be convex or not convex, and your sides can be crossed. Note 2: The angles are in counterclockwise. a) Find the length of the segment $A_nA_1$ for a brazilian figure with $n=5$. b) Find the length of the segment $A_nA_1$ for a brazilian figure with $n\equiv 1$ (mod $4$).