What is minimum value of |a+bw+cw^2|? whrere w is cube root of unity?

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The minimum value of |a + bw + cw²|, where w is the cube root of unity and a, b, c are integers not all equal, is established as being less than or equal to a + b + c. The analysis reveals that |a + bw + cw²| can be bounded by the triangle inequality, leading to the conclusion that at least one value of |a + bw + cw²| will be smaller than the minimum value of a + b + c. The confusion arises from the assumption regarding the magnitude of |w²|, which equals 1.

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vkash
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a,b,c are integers not all equal and w is the cube root of unity then minimum value of |a+bw+cw2|(w is not equals one).

My answer
|a+bw+cw2|<=|a|+|bw|+|cw2|
|a|+|bw|+|cw2|=a+b+c.
so at lest one value of |a+bw+cw2| will smaller than the minimum value of a+b+c. for integers this minimum value is smallest integers you can think of.
But that's wrong.?why?
 
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Is |w2| = 1?
 

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