  We present a precise characterization of the TOI-561 planetary system
obtained by combining previously published data with TESS and CHEOPS
photometry, and a new set of $62$ HARPS-N radial velocities (RVs). Our joint
analysis confirms the presence of four transiting planets, namely TOI-561 b ($P
= 0.45$ d, $R = 1.42$ R$_\oplus$, $M = 2.0$ M$_\oplus$), c ($P = 10.78$ d, $R =
2.91$ R$_\oplus$, $M = 5.4$ M$_\oplus$), d ($P = 25.7$ d, $R = 2.82$
R$_\oplus$, $M = 13.2$ M$_\oplus$) and e ($P = 77$ d, $R = 2.55$ R$_\oplus$, $M
= 12.6$ M$_\oplus$). Moreover, we identify an additional, long-period signal
($>450$ d) in the RVs, which could be due to either an external planetary
companion or to stellar magnetic activity. The precise masses and radii
obtained for the four planets allowed us to conduct interior structure and
atmospheric escape modelling. TOI-561 b is confirmed to be the lowest density
($\rho_{\rm b} = 3.8 \pm 0.5$ g cm$^{-3}$) ultra-short period (USP) planet
known to date, and the low metallicity of the host star makes it consistent
with the general bulk density-stellar metallicity trend. According to our
interior structure modelling, planet b has basically no gas envelope, and it
could host a certain amount of water. In contrast, TOI-561 c, d, and e likely
retained an H/He envelope, in addition to a possibly large water layer. The
inferred planetary compositions suggest different atmospheric evolutionary
paths, with planets b and c having experienced significant gas loss, and
planets d and e showing an atmospheric content consistent with the original
one. The uniqueness of the USP planet, the presence of the long-period planet
TOI-561 e, and the complex architecture make this system an appealing target
for follow-up studies.
