For a positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$. For example, $f_{1}(123) = (1+2+3)^{2} = 36$. Define $f_{n+1}(k) = f_{1}(f_{n}(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim.

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