
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 two different unit vectors in a plane $\overrightarrow{m}=({x}_{1},{y}_{1})$, $\overrightarrow{n}=({x}_{2},{y}_{2})$, and $\overrightarrow{p}=(1,1)$, where the angles formed by $\overrightarrow{m}$ and $\overrightarrow{p}$, and $\overrightarrow{n}$ and $\overrightarrow{p}$ are both $\frac{π}{6}$, find $\overrightarrow{m} \cdot \overrightarrow{n}=$______; $\frac{{y}_{1}{y}_{2}}{{x}_{1}{x}_{2}}=\_\_\_\_\_\_$.

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