= Solution
For $Y\gg k^2$, the <first integral of a square-root reaction layer> reduces to $Y'\sim(2/\sqrt3)Y^{3/4}$. Integrating gives $Y^{1/4}\sim\xi/(2\sqrt3)$ after a possible translation, or
$$
\boxed{Y\sim\frac{\xi^4}{144}.}
$$
For the right profile $R$ this large-positive-$\xi$ limit matches a broader scale with $y=O(1)$. Since $y=\varepsilon^2R$ and $\xi=(x-x_*)/\varepsilon$, the overlap is
$$
y\sim\frac{(x-x_*)^4}{144\varepsilon^2},\qquad \varepsilon\ll x-x_*\ll\sqrt\varepsilon.
$$
Thus the translated $O(\varepsilon)$ layer lies between the outer plateau and the $O(\sqrt\varepsilon)$ boundary layer. The large-$Y$ primitive also gives $\Phi(\varepsilon^{-2})\sim2\sqrt3/\sqrt\varepsilon$, establishing the leading location $x_*=1-2\sqrt{3\varepsilon}$.
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