with $\Delta$($U-I$)$>0.85$ with a Gaussian function by means of least squares (orange transparent line in the top-left panel of Figure \[fig:cmds\]). Then, we randomly associated each star with a probability to belong to the metal-poor and metal-rich sample based on ratio between the value of the best-fit Gaussian and the histogram distribution corresponding to its $\Delta$($U-I$) value. In the following, we present results based on the selection from procedure I, while in Section \[sec:summary\] we compare the results from the procedures I, II and III to demonstrate that the conclusions of the paper do not depend on the selection criterion.

  -------------------- ---------------- ----------------------- --------------------- -------------------- ----------------------- --------------------- ---------------------- ---------------------- -------------------------- ----------------------------- --
  ID                   RA (J2000)[^1]   DEC (J2000)$^{\rm a}$   mass$^{\rm b}$        $ d_{\rm sun}$[^2]   $R_{\rm Gal}^{\rm a}$   $R_{\rm c}^{\rm b}$   $R_{\rm h }^{\rm b}$   $R_{\rm t }^{\rm a}$   $\log t_{\rm h}^{\rm b}$   $\bar{V}_{\rm LoS}^{\rm b}$   
                                                                \[$M_{\rm \odot}$\]   \[kpc\]              \[kpc\]                 \[kpc\]               \[arcmin\]             \[arcmin\]             \[yr\]                     \[km/s\]                      
  $\omega\,$Centauri   13 26 47.24      $-$47 28 46.5           $3.36\times 10^6$     5.2                  6.40                    2.37                  5.00                   48.4                   10.39                      234.28                        
  M22                  18 36 23.94      $-$23 54 17.1           $4.16\times 10^5$     3.2                  4.90                    1.33                  3.36                   31.9                   9.45                       $-$147.76                     
  -------------------- ---------------- ----------------------- --------------------- -------------------- ----------------------- --------------------- ---------------------- ---------------------- -------------------------- ----------------------------- --

Multiple populations with 