A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). The probability that the sequence of jumps ends on a vertical side of the square is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Let's think step by step and output the final answer within \boxed{}.
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