
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 an angle of $\frac{\pi}{3}$ between them, and the magnitude of $\overrightarrow{n}$ is a positive scalar multiple of the magnitude of $\overrightarrow{m}$, i.e., $|\overrightarrow{n}| = λ|\overrightarrow{m}| (λ > 0)$. The vector group $\overrightarrow{x_1}, \overrightarrow{x_2}, \overrightarrow{x_3}$ consists of one $\overrightarrow{m}$ and two $\overrightarrow{n}$'s, while the vector group $\overrightarrow{y_1}, \overrightarrow{y_2}, \overrightarrow{y_3}$ consists of two $\overrightarrow{m}$'s and one $\overrightarrow{n}$. If the minimum possible value of $\overrightarrow{x_1} \cdot \overrightarrow{y_1} + \overrightarrow{x_2} \cdot \overrightarrow{y_2} + \overrightarrow{x_3} \cdot \overrightarrow{y_3}$ is $4\overrightarrow{m}^2$, then $λ =$ ___.

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