HMMT 2014 T6

Let $n$ be a positive integer. A sequence $\left(a_{0}, \ldots, a_{n}\right)$ of integers is acceptable if it satisfies the following conditions: (a) $0=\left|a_{0}\right|<\left|a_{1}\right|<\cdots<\left|a_{n-1}\right|<\left|a_{n}\right|$. (b) The sets $\left\{\left|a_{1}-a_{0}\right|,\left|a_{2}-a_{1}\right|, \ldots,\left|a_{n-1}-a_{n-2}\right|,\left|a_{n}-a_{n-1}\right|\right\}$ and $\left\{1,3,9, \ldots, 3^{n-1}\right\}$ are equal.

Prove that the number of acceptable sequences of integers is $(n+1)$ !.