
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 \( PQR \) and \( PTU \) are such that \( Q \) is the midpoint of \( \overline{TU} \). Also, \( PQ = TU = 2 \), \( QR = 8 \), \( PR = 2\sqrt{17} \), and
\[
\overrightarrow{PQ} \cdot \overrightarrow{PT} + \overrightarrow{PR} \cdot \overrightarrow{PU} = 4.
\]
Find the cosine of the angle between vectors \( \overrightarrow{TU} \) and \( \overrightarrow{QR} \).

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