Provide the electronic configurations of lanthanoids $(4f)$ series elements $(Ln)$ and their ions $Ln^{2+}$ and $Ln^{3+}$,and state their characteristics.

(A) The electronic configurations of the $15$ elements from Lanthanum $(La, Z=57)$ to Lutetium $(Lu, Z=71)$ and their $+2$ and $+3$ ions are summarized in the table below. Only electrons outside the $[Xe]$ core are shown.
| Atomic Number | Name | Symbol | $Ln$ | $Ln^{2+}$ | $Ln^{3+}$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $57$ | Lanthanum | $La$ | $5d^1 6s^2$ | $5d^1$ | $4f^0$ |
| $58$ | Cerium | $Ce$ | $4f^1 5d^1 6s^2$ | $4f^2$ | $4f^1$ |
| $59$ | Praseodymium | $Pr$ | $4f^3 6s^2$ | $4f^3$ | $4f^2$ |
| $60$ | Neodymium | $Nd$ | $4f^4 6s^2$ | $4f^4$ | $4f^3$ |
| $61$ | Promethium | $Pm$ | $4f^5 6s^2$ | $4f^5$ | $4f^4$ |
| $62$ | Samarium | $Sm$ | $4f^6 6s^2$ | $4f^6$ | $4f^5$ |
| $63$ | Europium | $Eu$ | $4f^7 6s^2$ | $4f^7$ | $4f^6$ |
| $64$ | Gadolinium | $Gd$ | $4f^7 5d^1 6s^2$ | $4f^7 5d^1$ | $4f^7$ |
| $65$ | Terbium | $Tb$ | $4f^9 6s^2$ | $4f^9$ | $4f^8$ |
| $66$ | Dysprosium | $Dy$ | $4f^{10} 6s^2$ | $4f^{10}$ | $4f^9$ |
| $67$ | Holmium | $Ho$ | $4f^{11} 6s^2$ | $4f^{11}$ | $4f^{10}$ |
| $68$ | Erbium | $Er$ | $4f^{12} 6s^2$ | $4f^{12}$ | $4f^{11}$ |
| $69$ | Thulium | $Tm$ | $4f^{13} 6s^2$ | $4f^{13}$ | $4f^{12}$ |
| $70$ | Ytterbium | $Yb$ | $4f^{14} 6s^2$ | $4f^{14}$ | $4f^{13}$ |
| $71$ | Lutetium | $Lu$ | $4f^{14} 5d^1 6s^2$ | $4f^{14} 5d^1$ | $4f^{14}$ |
Characteristics:
$1$. All lanthanoids have a $6s^2$ configuration in their neutral state.
$2$. $Gd$ and $Eu$ exhibit stable half-filled $4f^7$ subshells.
$3$. $Lu$ and $Yb$ exhibit stable fully-filled $4f^{14}$ subshells.