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Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.

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What positive integer value of \(M \) satisfies the following system of equations:

\[ \begin{cases} \sqrt{M} + \sqrt{N} = 25 \\ M - N = 175 ? \\ \end{cases} \]

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