2007 Fermat Problem 21

Five consecutive positive integers have the property that the sum of the second, third and fourth is a perfect square, while the sum of all five is a perfect cube. If m is the third of these five integers, then the minimum possible value of m satisfies

\textbf{(A)}\ m \le 200\quad \textbf{(B)}\ 200 < m \le 400\quad \textbf{(C)}\ 400 < m \le 600\quad \textbf{(D)}\ 600 < m \le 800\quad \textbf{(E)}\ m > 800

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