Let $f(x)$ be the function such that $f(x) > 0$ for $x \geq 0$ and $\{f(x)\}^{2006} = \int_{0}^{x} f(t) \, dt + 1.$

Find the value of $\{f(2006)\}^{2005}.$

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