= Solution
The beam rigidity is $B\rho\simeq p/(0.2998\ \mathrm{GeV}/(c\,\mathrm{T\,m}))\simeq1.67\times10^5$ T m. Equal opposing strengths have $f=B\rho/(GL_q)$, and weak separated-doublet net power is $1/F\simeq L/f^2$. Taking $F\sim L^*=50$ m gives the design family
$$
\boxed{L_q^2L\sim\frac{(B\rho/G)^2}{L^*}\simeq6.2\times10^3\ \mathrm{m^3}.}
$$
There is not a unique pair $(L_q,L)$ without a spacing/layout choice. A compact estimate $L\sim L_q$ gives about 18 m for each, showing immediately that neither is extremely small compared with 50 m and the advertised asymptotic approximation is only rough. For example $L=30$ m gives $L_q\sim14$ m in the same estimate.
An equal-strength doublet also has exact parallel-input focal distances $f^2/L-f$ and $f^2/L+f$: it is astigmatic, so a single exact common 50 m focus cannot be imposed without specifying incoming optical functions or allowing unequal strengths. The family above is a scale estimate of net focusing, not a complete collider final-focus design.
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