
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 points $A$ and $B$ move on the circle ${C}_{1}:(x-2)^{2}+{y}^{2}=3$, with $|AB|=2\sqrt{2}$, and point $P$ is the midpoint of $AB$. Point $Q$ moves on the circle ${C}_{2}:(x+2)^{2}+{y}^{2}=1$. Find the minimum value of $|PQ|$.

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