
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 $\overrightarrow{a} = (1, 2, 3)$ and $\overrightarrow{b} = (1, -1, 1)$, consider the set of vectors $A = \{\overrightarrow{x} | \overrightarrow{x} = \overrightarrow{a} + k\overrightarrow{b}, k \in \mathbb{Z}\}$. Find the minimum value of the magnitude $|\overrightarrow{x}|$.

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