Mr. 11 is back to the game!

Number Theory Level 5

If \(x,y,z \in \mathbb{Z},\) \(-100 \leq x,y,z \leq 100\) and \(x,y,z\) satisfy


then the number of ordered triples \((x,y,z)\) is \(n\).

If the smallest positive integer \(k\) satisfies the congruence \(n\cdot k \equiv 4 \pmod{11},\) what is the value of \(n+k\) ?

This problem is a part of 11≡ awesome (mod remainders).

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