
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 $O$ be the circumcenter of the acute triangle $\triangle ABC$, with $AB = 6$ and $AC = 10$. If $\overrightarrow{AO} = x \overrightarrow{AB} + y \overrightarrow{AC}$, and $2x + 10y = 5$, find $\cos \angle BAC = \quad$.

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