I am floored again

Let \( m \) and \( n\) be positive integers such that \(\textrm{gcd}(m,n)=1\). Further suppose that \(m\) is even and \(n\) is odd. Then what is the value of: \[ \frac{1}{2n} + \sum_{k=1}^{n-1} \left [ (-1)^{ \lfloor\frac{mk}{n}\rfloor} \left\{ \frac{mk}{n}\right\} \right] \]

Details and Assumptions:

  • \( \left\{\frac{mk}{n}\right\}\) denotes the fractional part of \(\frac{mk}{n}\).

  • \( \lfloor x \rfloor\) denotes the greatest integer \( \leq x \).

Remark:This problem is taken from a previous I.M.O Team Selection Test.

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