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.