Analytic regularity of CR maps into spheres

Let $M$ be a connected real-analytic hypersurface in ${\text{C}}^N$ and $\mathrm{\S }$ the unit real sphere in ${\text{C}}^{N^{\mathrm{\prime }}}$, $N^{\mathrm{\prime }}>N\ge 2$. Assume that $M$ does not contain any complex-analytic hypersurface of ${\text{C}}^N$ and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\text{\hspace{0.17em}⁣}:M\to \mathrm{\S }$ of class $C^{N^{\mathrm{\prime }}-N+1}$ extends holomorphically to a neighborhood of $M$ in ${\text{C}}^N$.