= Solution
With the negative exponential convention of the specified <quantum Fourier transform>, the amplitude of computational outcome $k$ is
$$
\langle k|U|\psi_m\rangle
=\frac1N\sum_{l=0}^{N-1}e^{2\pi i(m-k)l/N}.
$$
If $k=m$, every summand is one. Otherwise this finite <geometric series> has ratio $q\ne1$ with $q^N=1$, so it equals $(1-q^N)/(1-q)=0$. Thus
$$
\boxed{U|\psi_m\rangle=|m\rangle.}
$$
The <Born rule> gives outcome $m$ with probability one when performing a <quantum measurement in the computational basis>. Recover the promised phase as $\phi_m=2\pi m/N$ modulo $2\pi$. Using the opposite Fourier sign would instead return the label $-m\bmod N$, so the sign convention matters.
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