
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 any two non-zero planar vectors $\overrightarrow{α}$ and $\overrightarrow{β}$, define $\overrightarrow{α}○\overrightarrow{β}=\frac{\overrightarrow{α}·\overrightarrow{β}}{\overrightarrow{β}·\overrightarrow{β}}$. If two non-zero planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy that the angle $θ$ between $\overrightarrow{a}$ and $\overrightarrow{b}$ is in the interval $(π/6,π/2)$, and both $\overrightarrow{a}○\overrightarrow{b}$ and $\overrightarrow{b}○\overrightarrow{a}$ belong to the set ${n/2|n∈Z}$, calculate $\overrightarrow{a}○\overrightarrow{b}$.

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