Vector Ratio

Algebra Level 3

In the $$xyz$$-plane above, let $$O=(0,0,0)$$ be the point of origin, $$P=(a,b,c)$$, and $$Q=(a+b, b+c, c+a)$$, where $$a,b,c$$ are non-zero real numbers.

What is the maximum value of the ratio $$\frac{OQ}{OP}?$$

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