II. (50 points) If there exist real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying
$$
\begin{array}{l}
a_{n} \leqslant a_{n-1} \leqslant \cdots \leqslant a_{1}<2, \\
a_{1}+a_{2}+\cdots+a_{n} \geqslant n, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} \geqslant n^{2},
\end{array}
$$

Find all possible values of the positive integer $n$.