
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
Assume a point $P(x,y)$ is on the graph of the function $y=-\frac{1}{2}x^2+3\ln x (x > 0)$, and a point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$. The minimum value of $(x-m)^2+(y-n)^2$ is \_\_\_\_\_\_.

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