Why the Helmholtz free energy is minimized when states obey the Boltzmann distribution?

The Helmholtz free energy is defined by the difference between the internal energy and the entropy of the system,

$$F_{T} = U_{T} + kTH_{T},$$

where $U_{T} = sum_s P_{T}(s)E(s)$ is the internal energy of a state $s$ and $H_{T} = -sum_s P_{T}(s) log P_{T}(s)$ is the entropy. Why $P_{T}(s)$ is the Boltzmann distribution when $F$ is minimized?

I try to minimize $F_{T}$ w.r.t. $P_{T}(s)$ but fail to obtain the Boltzmann distribution.

$$begin{aligned}
& frac{d F_{T}}{d P_{T}(s)}= E_{s}-k T left[log P_{T}(s) + 1right] = 0
& Rightarrow E_{s} = k T left[log P_{T}(s) + 1right]
& Rightarrow P_{T}(s) = exp(frac{E_{s}}{k T}-1) quad s.t. quad sum_s P_{T}(s) = 1
& Rightarrow P_{T}(s) = exp left( frac{E_{s}}{k T}- sum_s exp(frac{E_{s}}{k T}-1) right)
& Rightarrow P_{T}(s) = frac{exp (frac{E_{s}}{k T})}{sum_s exp(frac{E_{s}}{k T}-1) }
end{aligned}$$