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Algebra Level 2

\[\large{\begin{cases} a + \frac 1b = \frac73 \\ b + \frac1c = 4 \\ c + \frac1a = 1 \end{cases}}\]

If \(a,b,\) and \(c\) are real numbers that satisfy the system of equations above, find the value of \(abc\).

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