
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 real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$.

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