For a positive integer $N$, let $f_N^{1}$ be the function defined by
$$f_N(x)=\sum_{n=0}^N\frac{N+1/2-n}{(N+1)(2n+1)}\sin((2n+1)x).$$
Determine the smallest constant $M$ such that $f_N(x)\leq M$
 for all $N$ and all real $x.$ The original answer is in $\frac{m\pi}{n}$ format, please give the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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