Log of arithmetic progression

Algebra Level 4

If \(x\), \(y\), \(z\) and \(v\) are 4 integers in an arithmetic progression and

\[1000 ^ {\log (x)} + 1000 ^ {\log (y)} + 1000 ^ {\log (z)} = 1000 ^ {\log (v)}. \]

Find the minimum possible value of \(x + y + z + v\).

Clarification: All the logarithms are in base 10.

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