
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 real numbers \(a, b, c, d\), the following equations hold:

\[ 4a^{2} + b^{2} - 8b + 12 = 0 \]

\[ c^{2} - 8c + 4d^{2} + 12 = 0 \]

What is the minimum value of the expression \((a - c)^{2} + (b - d)^{2}\)?

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