Freaky Factorials

Find the sum of all integers $$m$$ that are less than $$1000$$ and equal to $$\sqrt{n!+1}$$ for some positive integer $$n$$.

Details and assumptions

The number $$n!$$, read as n factorial, is equal to the product of all positive integers less than or equal to $$n$$. For example, $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$$.

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