
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 $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, and the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is $\frac{\pi}{3}$. Let $\overrightarrow{m}=\overrightarrow{a}+t\overrightarrow{b}$ and $\overrightarrow{n}=t\overrightarrow{a}+2\overrightarrow{b}$.  
$(1)$ Find the value of $\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(2)$ Define $f(t)=\overrightarrow{m}\cdot\overrightarrow{n}$. Find the expression for $f(t)$.  
$(3)$ Define $g(t)=\frac{f(t)}{t}$. Find the range of $g(t)$ on the interval $[1,3]$.

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