Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$.  Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$.

(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

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