
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 the trapezoid $ABCD$, it is known that $AB$ is parallel to $CD$, $B=4$, $BC=a$, $\angle ABC=60^{\circ}$. Points $E$ and $F$ are located on the extensions of $BC$ and $CD$ respectively, with $\overrightarrow{BE}=λ \overrightarrow{BC}$ and $\overrightarrow{DF}= \frac{1}{9λ} \overrightarrow{DC}$. Determine the value of $λ$ when $\overrightarrow{AE} \cdot \overrightarrow{AF}$ is minimized, and find this minimum value.

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