Repeated Roots

Calculus Level 2

There exists an expression with infinite nested roots that evaluates to an integer:

\[3=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}}}.\]

How many integer values of \(x\) between 1 and 1000 (inclusive) are there such that \[\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}}\] is also an integer?

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