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Prompt: Let $a, b,$ and $c$ be complex numbers such that $a+b+c = 3$ and $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$. Let $\zeta$ be a non-real cube root of unity and let $w = a + b\zeta + c\zeta^2$. Evaluate the value of $|w| \cdot \sqrt{3}$.

Since $3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc = 24$, we can substitute this into the expression for $w^3$: $$w^3 = a^3 + b^3 + c^3 + 24\zeta + 0\zeta^2.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

However, we can simplify this further by noticing that $1 + \zeta + \zeta^2 = 0$, so $\zeta^2 = -1 - \zeta$. Substituting this into the expression for $w^3$, we get: $$w^3 = a^3 + b^3 + c^3 + 24\zeta - 0(1 + \zeta).$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Since $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$, we can substitute this into the expression for $w^3$: $$w^3 = (a+b+c)^3 + 24\zeta - 24.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Now, we can substitute $a+b+c = 3$ into the expression for $w^3$: $$w^3 = 3^3 + 24\zeta - 24 = 27 + 24\zeta - 24.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Simplifying the expression for $w^3$, we get: $$w^3 = 3 + 24\zeta.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Taking the absolute value of $w^3$, we get: $$|w^3| = |3 + 24\zeta|.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Since $\zeta$ is a non-real cube root of unity, we know that $|\zeta| = 1$. Therefore, we can simplify the expression for $|w^3|$ as follows: $$|w^3| = \sqrt{3^2 + 24^2} = \sqrt{9 + 576} = \sqrt{585}.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Since $|w^3| = |w|^3$, we can take the cube root of both sides to get: $$|w| = \sqrt[3]{\sqrt{585}}.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Simplifying the expression for $|w|$, we get: $$|w| = \sqrt[3]{\sqrt{9 \cdot 65}} = \sqrt[3]{3 \sqrt{65}}.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Finally, we can multiply $|w|$ by $\sqrt{3}$ to get: $$|w| \cdot \sqrt{3} = \sqrt[3]{3 \sqrt{65}} \cdot \sqrt{3} = \sqrt[3]{3^3 \sqrt{65}} = \boxed{\sqrt{65}}.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate