A line initially \(1\) inch long grows according to the following law, where the first term is the initial length:
\[
1 + \frac{1}{4}\sqrt{2} + \frac{1}{4} + \frac{1}{16}\sqrt{2} + \frac{1}{16} + \frac{1}{64}\sqrt{2} + \frac{1}{64} + \cdots.
\]
If the growth process continues forever, what is the limit of the length of the line? The original answer is in the format \(\frac{k}{m}(n + \sqrt{p})\), please give the value of k + m + n + p.

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