
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 that the graph of function $g(x)$ is symmetric with respect to the line $y=x$ to the graph of function $f(x)=x^{2}$ for $x\in [0, +\infty)$, the graph of function $g(x)$ is shifted $2$ units to the right and $2$ units down to obtain the graph of function $h(x)$. If $P$ and $Q$ are two moving points on the graphs of functions $f(x)$ and $h(x)$ respectively, then the minimum distance between these two points is ______.

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