The arithmetic mean, geometric mean, and harmonic mean of \(a,b,c\) are 8, 5, 3, respectively. What is the value of \( a^2 + b^2 + c^2 \)?

This problem is posed by Matt.

**Details and Assumptions:**

The arithmetic mean, geometric mean, and harmonic mean of \(n\) numbers \(a_1, a_2, \ldots, a_n\) are (respectively)

\[ \frac {\sum_{i=1}^{n} a_n}{n},\quad \sqrt[n]{\prod_{i=1}^{n} a_n},\quad \left( \frac{n} { \sum_{i=1}^{n} \frac{1}{a_n} } \right) . \]

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