
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 plane vectors $a, b,$ and $c$ such that:
$$
\begin{array}{l}
|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=2, \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}.
\end{array}
$$

If $0 \leqslant x \leqslant \frac{1}{2} \leqslant y \leqslant 1$, then find the minimum value of 
$$
|x(\boldsymbol{a}-\boldsymbol{c})+y(\boldsymbol{b}-\boldsymbol{c})+\boldsymbol{c}|.
$$

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