
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 $\overrightarrow{a\_k} = (\cos \frac{k\pi}{6}, \sin \frac{k\pi}{6} + \cos \frac{k\pi}{6})$ for $k \in \mathbb{Z}$, find the dot product of $\overrightarrow{a_{2015}}$ and $\overrightarrow{a_{2016}}$, denoted as $\overrightarrow{a_{2015}} \cdot \overrightarrow{a_{2016}}$.

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