
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 that $\overrightarrow{e_1}, \overrightarrow{e_2}$ are unit vectors, and the non-zero vector $\overrightarrow{b}=x \overrightarrow{e_1}+y \overrightarrow{e_2},x,y\in\mathbb{R}$, and the angle between $\overrightarrow{e_1}, \overrightarrow{e_2}$ is $\frac{\pi}{6}$, determine the maximum value of $\frac{|x|}{|\overrightarrow{b}|}$.

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