
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 $\angle ABC=60^{\circ}$. Points $E$ and $F$ move along segments $BC$ and $DC$ respectively, and $\vec{BE}=\lambda \vec{BC}$, $\vec{DF}= \frac{1}{4\lambda} \vec{DC}$. Find the minimum value of $\vec{AE} \cdot \vec{AF}$.

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