Find all $\alpha_i, 1\le i \le n$ and $\beta_j, 1 \le j \le n$, $\alpha_i, \beta_j \in \mathbb{N} $ satisfying the following:
(1) $\alpha_i \ge \alpha_{i+1}$ for $1 \le i \le n-1$ and $\beta_j \ge \beta_{j+1}$ for $1 \le j \le n-1$.
(2) $\Sigma \alpha_i = \Sigma \beta_j$.
(3) the sets $\lbrace \alpha_i +i-1 \rbrace_1^n$ and $\lbrace \beta_j +j-1 \rbrace_1^n$ are same.