
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 three vectors in space, $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$, where $\overrightarrow{a}$ and $\overrightarrow{b}$ are two mutually perpendicular unit vectors, and vector $\overrightarrow{c}$ satisfies $|\overrightarrow{c}| = 3$, $\overrightarrow{c} \cdot \overrightarrow{a} = 2$, and $\overrightarrow{c} \cdot \overrightarrow{b} = 1$. Find the minimum value of $|\overrightarrow{c} - x\overrightarrow{a} - y\overrightarrow{b}|$ for any real numbers $x$ and $y$.

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