
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 $APQ$ are such that $P$ is the midpoint of $\overline{AQ}$. Also, $AB = AQ = 1$, $BC = 8$, $CA = \sqrt{73}$, and
\[
\overrightarrow{AB} \cdot \overrightarrow{AP} + \overrightarrow{AC} \cdot \overrightarrow{AQ} = 3.
\]
Find the cosine of the angle between vectors $\overrightarrow{AQ}$ and $\overrightarrow{BC}$.

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