
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 the variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$.

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