
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
In $\triangle ABC$, $BC=2$. For any real number $t$, if $|t \overrightarrow{AB}+(1-t) \overrightarrow{AC}|\geqslant |t_{0} \overrightarrow{AB}+(l-t_{0}) \overrightarrow{AC}|=3 (t_{0}\in R)$, then the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is $\_\_\_\_\_\_$, and $t_{0}=$ $\_\_\_\_\_\_$.

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