Floor Function Power!

\[ \large \sum_{k=0}^{mn-1} \exp_{-1} \Bigg ( \left \lfloor \frac km \right \rfloor + \left \lfloor \frac kn \right \rfloor \Bigg) \]

Let \(m\) and \(n\) be relatively prime positive integers such that \(mn\) is odd. Then what is the value of summation above.

Note the notation, \(\exp_A B = A^B \).

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