A $24\mathrm{.}5$ pm x$-$ray photon scatters off a free electron at $A($see the figure below$),$ producing a photon of wavelength ${}^{\text{'}}$ traveling at an angle $=45\mathrm{.}0$ relative to the first photon's direction. This second photon scatters off another free electron at $B,$ producing a photon with wavelength $"$ and moving in a direction directly opposite the first photon. Determine the wavelengths ${}^{\text{'}}$ and ${}^{\text{'}\text{'}},$ both in pm$.$

$($a$){}^{\text{'}}$

$$ pm

$($b$){}^{\text{'}\text{'}}$

$$ pm