
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 $\overrightarrow{\beta}$, define $\overrightarrow{\alpha}○\overrightarrow{\beta}=\dfrac{\overrightarrow{\alpha}⋅\overrightarrow{\beta}}{\overrightarrow{\beta}⋅\overrightarrow{\beta}}$. If plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|\geqslant |\overrightarrow{b}| > 0$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta\in(0,\dfrac{\pi}{4})$, and both $\overrightarrow{a}○\overrightarrow{b}$ and $\overrightarrow{b}○\overrightarrow{a}$ are in the set $\{\dfrac{n}{2}|n\in\mathbb{Z}\}$, find the value of $\overrightarrow{a}○\overrightarrow{b}$.

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