
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 non-zero vectors $\overrightarrow{m}, \overrightarrow{n}$ with an angle of $\frac{\pi}{3}$ between them, and satisfying $|\overrightarrow{n}| = \lambda|\overrightarrow{m}|$ ($\lambda > 0$), consider two sets of vectors $\overrightarrow{x_{1}}, \overrightarrow{x_{2}}, \overrightarrow{x_{3}}$ composed of one $\overrightarrow{m}$ and two $\overrightarrow{n}$ vectors and $\overrightarrow{y_{1}}, \overrightarrow{y_{2}}, \overrightarrow{y_{3}}$ composed of two $\overrightarrow{m}$ and one $\overrightarrow{n}$ vectors. 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 $4|\overrightarrow{m}|^2$, find the value of $\lambda$.

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