Given a positive integer $n \geqslant 2$, real numbers $a_{1}, a_{2}, \cdots$, $a_{n}$ satisfy $\sum_{i=1}^{n}\left|a_{i}\right|+\left|\sum_{i=1}^{n} a_{i}\right|=1$. Find the minimum and maximum values of $\sum_{i=1}^{n} a_{i}^{2}$.
On the one hand, by the Cauchy-Schwarz inequality, we have
$$
\begin{array}{l}
\sqrt{n \sum_{i=1}^{n} a_{i}^{2}} \geqslant \sum_{i=1}^{n}\left|a_{i}\right|, \\
\sqrt{n \sum_{i=1}^{n} a_{i}^{2}} \geqslant\left|\sum_{i=1}^{n} a_{i}\right| .
\end{array}
$$

Adding the two inequalities, we get
$$
2 \sqrt{n \sum_{i=1}^{n} a_{i}^{2}} \geqslant 1 \Rightarrow \sum_{i=1}^{n} a_{i}^{2} \geqslant \frac{1}{4 n} \text {. }
$$

Equality holds when \( a_{1}=a_{2}=\cdots=a_{n}=\frac{1}{2 n} \).
On the other hand, for \( 1 \leqslant j \leqslant n \), by the triangle inequality, we have
$$
\begin{array}{l}
1=\sum_{i=1}^{n}\left|a_{i}\right|+\left|\sum_{i=1}^{n} a_{i}\right| \\
\geqslant\left|\sum_{i \neq j}\left(a_{i}-a_{j}\right)-\sum_{i=1}^{n} a_{i}\right|=2\left|a_{j}\right| \\
\Rightarrow\left|a_{j}\right| \leqslant \frac{1}{2} \\
\Rightarrow \sum_{i=1}^{n} a_{i}^{2} \leqslant \frac{1}{2} \sum_{i=1}^{n}\left|a_{i}\right| \leqslant \frac{1}{2} .
\end{array}
$$

Equality holds when \( a_{1}=\frac{1}{2}, a_{2}=-\frac{1}{2}, a_{3}=\cdots=a_{n}=0 \).
In summary, the minimum value of \( \sum_{i=1}^{n} a_{i}^{2} \) is \( \frac{1}{4 n} \), and the maximum value is \( \frac{1}{2} \).