
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
For any two non-zero plane vectors $\overrightarrow \alpha$ and $\overrightarrow \beta$, a new operation $\odot$ is defined as $\overrightarrow \alpha ⊙ \overrightarrow \beta = \frac{\overrightarrow \alpha • \overrightarrow \beta}{\overrightarrow \beta • \overrightarrow \beta}$. Given non-zero plane vectors $\overrightarrow a$ and $\overrightarrow b$ such that $\overrightarrow a ⊙ \overrightarrow b$ and $\overrightarrow b ⊙ \overrightarrow a$ are both in the set $\{x|x=\frac{{\sqrt{3}k}}{3}, k∈{Z}\}$, and $|\overrightarrow a| \geq |\overrightarrow b|$. Let the angle between $\overrightarrow a$ and $\overrightarrow b$ be $θ∈(\frac{π}{6},\frac{π}{4})$, then $(\overrightarrow a ⊙ \overrightarrow b) \sin θ =$ ____.

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