
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 two non-zero vectors $\overrightarrow{m}$ and $\overrightarrow{n}$, with $|\overrightarrow{m}| = 2$ and $|\overrightarrow{m} + 2\overrightarrow{n}| = 2$, find the maximum value of $|\overrightarrow{n}| + |2\overrightarrow{m} + \overrightarrow{n}|$.

A) $4\sqrt{2}$
B) $3\sqrt{3}$
C) $\frac{7\sqrt{3}}{2}$
D) $\frac{8\sqrt{3}}{3}$

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