Fermat's Last Theorem

Andrew Wiles, pictured above, was recently awarded the 2016 Abel Prize for his 1994 proof of Fermat's Last Theorem.

Suppose \(n=2016\) and \(|x|,|y|,|z|\) are all less than 10. How many solutions in integers are there to the equation on the chalkboard?

Note: A solution here refers to an ordered triple \((x,y,z).\)

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