
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
Given the plane vectors $\overrightarrow{a}=(x_{1},y_{1})$ and $\overrightarrow{b}=(x_{2},y_{2})$, then $\overrightarrow{a} \cdot \overrightarrow{b}=x_{1}x_{2}+y_{1}y_{2}$; for space vectors $\overrightarrow{a}=(x_{1},y_{1},z_{1})$ and $\overrightarrow{b}=(x_{2},y_{2},z_{2})$, then $\overrightarrow{a} \cdot \overrightarrow{b}=x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}$. By extending this to $n$-dimensional vectors: $\overrightarrow{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\overrightarrow{b}=(b_{1},b_{2},\ldots,b_{n})$, then $\overrightarrow{a} \cdot \overrightarrow{b}=$ ______.

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