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There exists an expression with infinite nested roots that evaluates to an integer:
3=6+6+6+6+6+⋯.3=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}}}.3=6+6+6+6+6+⋯.
How many integer values of xxx between 1 and 1000 (inclusive) are there such that x+x+x+x+x+⋯\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}}x+x+x+x+x+⋯ is also an integer?
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