
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in a plane that satisfy $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=\sqrt{3}$, and the minimum value of $|x\overrightarrow{a}+(1-2x)\overrightarrow{b}|$ ($x\in\mathbb{R}$) is $\frac{\sqrt{3}}{2}$, find the minimum value of $|\overrightarrow{a}+y\overrightarrow{b}|$ ($y\in\mathbb{R}$).

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