
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
We call an ordered real array of $n$ elements $(x_{1}$,$x_{2}$,$\cdots $,$x_{n})$ with $n\in N^{*}$ a $n$-dimensional vector, and the norm of this vector is defined as $|x_{1}|+|x_{2}|+\cdots +|x_{n}|$. Given a $n$-dimensional vector $\overrightarrow{a}=({x}_{1},{x}_{2},⋯,{x}_{n})$, where $x_{i}\in \{-1,0,1\}$ for $i=1,2,\cdots,n$, let $A_{n}$ denote the number of vectors $\overrightarrow{a}$ with odd norm. Find $A_{3}=$______ and $A_{n}=$______, where $n\in N^{*}$.

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