
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
If point $P$ is located within the plane region defined by $\begin{cases}\begin{matrix}2x-y+2\geqslant 0 \\ x-2y+1\leqslant 0\\\end{matrix} \\ x+y-2\leqslant 0\end{cases}$, and point $Q$ is on the curve defined by $x^{2}+(y+2)^{2}=1$, determine the minimum value of $|PQ|$.

A) $\sqrt{5} -1$
B) $\frac{4}{ \sqrt{5}} -1$
C) $2\sqrt{2} -1$
D) $\sqrt{2} -1$

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