
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 $PST$ are such that $Q$ is the midpoint of $\overline{ST}.$ Also, $PQ = ST = 1,$ $QR = 8,$ $PR = \sqrt{65},$ and
\[\overrightarrow{PQ} \cdot \overrightarrow{PS} + \overrightarrow{PR} \cdot \overrightarrow{PT} = 2.\] Find the cosine of the angle between vectors $\overrightarrow{ST}$ and $\overrightarrow{QR}.$

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