= Solution
The fast scale resolves the rapid complex buildup that is compressed into a thin <initial layer> on the <substrate>-depletion scale. The outer scale follows depletion over a much longer time and cannot satisfy $c(0)=0$ by itself.
In the overlap $1\ll\tau_f\ll\epsilon^{-1}$, the <inner expansion>[inner solution] tends to $(s,c)=(1,1/(1+K))$. The <outer solution> as $\tau_s\downarrow0$ has the same limit. Subtracting this common part produces a leading <composite asymptotic expansion>[composite approximation],
$$
\boxed{s_{\rm comp}=s_{\rm out}(\tau_s),\quad
c_{\rm comp}=\frac{s_{\rm out}(\tau_s)}{s_{\rm out}(\tau_s)+K}-\frac{e^{-(1+K)\tau_s/\epsilon}}{1+K}.}
$$
It satisfies the leading initial values and tends to the outer quasi-steady solution after the <initial layer>. Corrections of order $\epsilon$ describe the small <substrate> loss during complex buildup.
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