
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
Define a mapping rule where a point $P(m,n)$ corresponds to a point $Q(-\sqrt{n}, -\frac{\sqrt{m}}{2})$, where $m \geq 0$ and $n \geq 0$. According to mapping rule $f$, as point $P$ moves along segment $AB$ from point $A(4,0)$ to point $B(0,4)$, the corresponding trajectory of point $Q$ can be obtained, denoted by curve $E$. Calculate the minimum distance between points on curve $E$ and segment $AB$.

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