In the city of $\textrm{Las Cobayas}$, the houses are arranged in a rectangular grid of $3$ rows and $n\geq 2$ columns, as illustrated in the figure. $\textrm{Mich}$ plans to move there and wants to tour the city to visit some of the houses in a way that he visits at least one house from each column and does not visit the same house more than once. During his tour, $\textrm{Mich}$ can move between adjacent houses, that is, after visiting a house, he can continue his journey by visiting one of the neighboring houses to the north, south, east, or west, which are at most four. The figure illustrates one $\textrm{Mich´s}$ position (circle), and the houses to which he can move (triangles). Let $f(n)$ be the number of ways $\textrm{Mich}$ can complete his tour starting from a house in the first column and ending at a house in the last column. Prove that $f(n)$ is odd. [asy][asy]size(200); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); fill(circle((0.5,2.5), 0.4), black); fill((0.1262,4.15)--(0.8738,4.15)--(0.5,4.7974)--cycle, black); fill((0.1262,0.15)--(0.8738,0.15)--(0.5,0.7974)--cycle, black); fill((2.1262,2.15)--(2.8738,2.15)--(2.5,2.7974)--cycle, black); fill(circle((6,0.5), 0.07), black); fill(circle((6.3,0.5), 0.07), black); fill(circle((6.6,0.5), 0.07), black); fill(circle((6,2.5), 0.07), black); fill(circle((6.3,2.5), 0.07), black); fill(circle((6.6,2.5), 0.07), black); fill(circle((6,4.5), 0.07), black); fill(circle((6.3,4.5), 0.07), black); fill(circle((6.6,4.5), 0.07), black); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((10,0)--(11,0)--(11,1)--(10,1)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((10,2)--(11,2)--(11,3)--(10,3)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((10,4)--(11,4)--(11,5)--(10,5)--cycle); draw((0,-0.2)--(0,-0.5)--(5.5,-0.5)--(5.5,-0.8)--(5.5,-0.5)--(11,-0.5)--(11,-0.5)--(11,-0.2)); label("$n$", (5.22,-1.15), dir(0), fontsize(10)); label("$\textrm{West}$", (-2,2.5), dir(0), fontsize(10)); label("$\textrm{East}$", (11.1,2.5), dir(0), fontsize(10)); label("$\textrm{North}$", (4.5,5.7), dir(0), fontsize(10)); label("$\textrm{South}$", (4.5,-2), dir(0), fontsize(10)); draw((0.5,2.5)--(2,2.5)--(1.8,2.7)--(2,2.5)--(1.8,2.3)); draw((0.5,2.5)--(0.5,4)--(0.3,3.7)--(0.5,4)--(0.7,3.7)); draw((0.5,2.5)--(0.5,1)--(0.3,1.3)--(0.5,1)--(0.7,1.3)); [/asy][/asy]