# Cyclic Cube Root

Let $$a,b,c$$ be integers satisfying the equations

$a+b+c+1=2014$

$\sum_{cyc}\sqrt[3]{a-b+1}=3$

Find $\text{min }\text{min}(a,b,c)$

Details and Assumptions

$$\text{min }\text{min}(a,b,c)$$ means the minimum of all possible minimums of $$a,b,c$$ where $$a,b,c$$ ranges over all possible solutions to the equations.

$$\displaystyle\sum_{cyc}\sqrt[3]{a-b+1}=\sqrt[3]{a-b+1}+\sqrt[3]{b-c+1}+\sqrt[3]{c-a+1}$$

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