
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
Given that vectors $\overrightarrow{e\_1}$ and $\overrightarrow{e\_2}$ are unit vectors in the same direction as the $x$-axis and $y$-axis respectively, vector $\overrightarrow{OA} = \overrightarrow{e\_1} + \overrightarrow{e\_2}$, and $\overrightarrow{OB} = 5\overrightarrow{e\_1} + 3\overrightarrow{e\_2}$. The directed line segment $\overrightarrow{AB}$ is rotated around point $A$ to the position of $\overrightarrow{AC}$ such that $\overrightarrow{AB} \perp \overrightarrow{AC}$. Determine the value of $\overrightarrow{OB} \cdot \overrightarrow{OC}$.

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