
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$ and $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{2}$, if $\overrightarrow{c} = x\overrightarrow{a} + y\overrightarrow{b}$, where $x > 0$, $y > 0$, and $x+y=2$, then the minimum value of $|\overrightarrow{c}|$ is \_\_\_\_\_\_.

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