1. Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of $1,2, \cdots, n$. It is known that $0, a_{1}, a_{1}+a_{2}, \cdots, a_{1}+a_{2}+\cdots+a_{n}$ when divided by $n+1$ have distinct remainders. Find all possible values of the positive integer $n$.
Prompt: In $0, a_{1}, a_{1}+a_{2}, \cdots, a_{1}+a_{2}+\cdots+a_{n}$, any two numbers are not congruent modulo $n+1$, especially,
$$
a_{1}+a_{2}+\cdots+a_{n}=\frac{n(n+1)}{2} \not \equiv 0(\bmod n+1) \text {. }
$$

Therefore, $n$ is odd.
For $n=2 k-1$, construct an example,
$$
\begin{array}{l}
a_{2 t-1}=2 t-1(1 \leqslant t \leqslant k) \\
a_{2 t}=2 k-2 t(1 \leqslant t \leqslant k-1)
\end{array}
$$