
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 the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is $\frac{2\pi}{3}$, and $|\overrightarrow{a}| = |\overrightarrow{b}| = 2$, and vector $\overrightarrow{c} = x\overrightarrow{a} + y\overrightarrow{b}$ ($x \in \mathbb{R}$ and $x \neq 0$, $y \in \mathbb{R}$), find the maximum value of $|\frac{|x|}{|\overrightarrow{c}|}$.

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