
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 $\overrightarrow{a}$ and $\overrightarrow{b}$, define $\overrightarrow{a} \otimes \overrightarrow{b} = \frac{|\overrightarrow{a}|}{|\overrightarrow{b}|}\cos \theta$, where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. If two non-zero planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy: ① $|\overrightarrow{a}| \geqslant |\overrightarrow{b}|$; ② the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$ is in $(0, \frac{\pi}{4})$; ③ both $\overrightarrow{a} \otimes \overrightarrow{b}$ and $\overrightarrow{b} \otimes \overrightarrow{a}$ are in the set $\{x|x = \frac{n}{2}, n \in \mathbb{N}\}$, then the value of $\overrightarrow{a} \otimes \overrightarrow{b}$ is $\_\_\_\_\_\_\_\_\_.$

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