
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 isosceles trapezoid $ABCD$, it is known that $AB\parallel DC$, $AB=2$, $BC=1$, and $\angle ABC=60^{\circ}$. Moving points $E$ and $F$ are on line segments $BC$ and $DC$ respectively, and $\overrightarrow{BE}=\lambda \overrightarrow{BC}$, $\overrightarrow{DF}= \frac{1}{9\lambda} \overrightarrow{DC}$. The minimum value of $\overrightarrow{AE}\cdot \overrightarrow{AF}$ is \_\_\_\_\_\_.

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