
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 variables $a$ and $b$ satisfy the equation $b=-\frac{1}{2}a^2 + 3\ln{a} (a > 0)$, and that point $Q(m, n)$ lies on the line $y = 2x + \frac{1}{2}$, find the minimum value of $(a - m)^2 + (b - n)^2$.

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