
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
Let $D$ and $E$ be the midpoints of line segments $AB$ and $AC$, respectively, and $\overrightarrow{BE} \cdot \overrightarrow{CD} = 0$. Let $\alpha$ be the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$. The minimum value of $\cos 2\alpha$ is \_\_\_\_\_\_.

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