  We report a low-temperature specific heat study of high-quality single
crystals of the heavily hole doped superconductor
Ca$_{0.32}$Na$_{0.68}$Fe$_2$As$_2$. This compound exhibits bulk
superconductivity with a transition temperature $T_c \approx 34$\,K, which is
evident from the magnetization, transport, and specific heat measurements. The
zero field data manifests a significant electronic specific heat in the normal
state with a Sommerfeld coefficient $\gamma \approx 53$ mJ/mol K$^{2}$. Using a
multi-band Eliashberg analysis, we demonstrate that the dependence of the zero
field specific heat in the superconducting state is well described by a
three-band model with an unconventional s$_\pm$ pairing symmetry and gap
magnitudes $\Delta_i$ of approximately 2.35, 7.48, and -7.50 meV. Our analysis
indicates a non-negligible attractive intraband coupling,which contributes
significantly to the relatively high value of $T_c$. The Fermi surface averaged
repulsive and attractive coupling strengths are of comparable size and outside
the strong coupling limit frequently adopted for describing high-$T_c$ iron
pnictide superconductors. We further infer a total mass renormalization of the
order of five, including the effects of correlations and electron-boson
interactions.
