
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
Triangles $ABC$ and $ADE$ are such that $B$ is on $\overline{DE}$ such that $DB = \frac{1}{3}DE$ and $BE = \frac{2}{3}DE$. Also, $AB = DE = 1$, $BC = 3$, $CA = \sqrt{5}$, and
\[
\overrightarrow{AB} \cdot \overrightarrow{AD} + \overrightarrow{AC} \cdot \overrightarrow{AE} = 1.
\]
Find the cosine of the angle between vectors $\overrightarrow{DE}$ and $\overrightarrow{BC}$.

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