Cyclic Cube Root

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

a+b+c+1=2014a+b+c+1=2014

cycab+13=3\sum_{cyc}\sqrt[3]{a-b+1}=3

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

Details and Assumptions

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

cycab+13=ab+13+bc+13+ca+13\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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