
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A sequence of $n$ ($n\geqslant 2$) distinct numbers $a\_1$, $a\_2$, ..., $a\_n$ is said to have an inversion if $1\leqslant i < j\leqslant n$ and $a\_j < a\_i$ (i.e., the later term $a\_j$ is smaller than the earlier term $a\_i$). The total number of inversions in a finite sequence is called the inversion count of that sequence. For example, the sequence 3, 2, 1 has an inversion count of 3, and the geometric sequence $1,- \frac {1}{2}, \frac {1}{4},- \frac {1}{8}$ has an inversion count of 4.

1. Calculate the inversion count of the sequence $a\_n=-2n+19$ ($1\leqslant n\leqslant 100,n\in\mathbb{N}^*$);
2. Calculate the inversion count of the sequence $a\_n= \begin{cases} ( \frac {1}{3})^{n},n\text{ is odd} \\ -\frac {n}{n+1},n\text{ is even}\end{cases}$ ($1\leqslant n\leqslant k,n\in\mathbb{N}^*$);
3. Given that the inversion count of the sequence $a\_1$, $a\_2$, ..., $a\_n$ is $a$, find the inversion count of $a\_n$, $a\_{n-1}$, ..., $a\_1$.

Present the answer in LaTex format: \boxed{Your answer}
user