
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 the vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ with an angle of $\frac{2\pi}{3}$ between them, and $|\overrightarrow{n}|=2|\overrightarrow{m}|$, if $(\lambda \overrightarrow{m}+ \overrightarrow{n})$ is perpendicular to $(\overrightarrow{m}-2 \overrightarrow{n})$, find the value of $\lambda$.

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