One, (15 points) Find the maximum value of the function
$$
y=\left[\sin \left(\frac{\pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right)\right] \sin \left(\frac{\pi}{3}+x\right)
$$

and the set of $x$ values at which the maximum value is attained.
$$
\begin{array}{l}
y=\left[\sin \left(\frac{\pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right)\right] \sin \left(\frac{\pi}{3}+x\right) \\
=\sqrt{2} \sin x\left(\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x\right) \\
=\frac{\sqrt{6}}{2} \sin x \cdot \cos x+\frac{\sqrt{2}}{2} \sin ^{2} x \\
=\frac{\sqrt{6}}{4} \sin 2 x+\frac{\sqrt{2}}{4}(1-\cos 2 x) \\
=\frac{\sqrt{2}}{2} \sin \left(2 x-\frac{\pi}{6}\right)+\frac{\sqrt{2}}{4} .
\end{array}
$$

Therefore, $y_{\max }=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{4}=\frac{3 \sqrt{2}}{4}$.
At this point, the set of $x$ values is $\left\{x \left\lvert\, x=k \pi+\frac{\pi}{3}\right., k \in \mathbf{Z}\right\}$.