
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 $(a,b)$ that satisfy the equation $\ln b + 1 + a - 3b = 0$, and the real numbers $(c,d)$ that satisfy the equation $2d - c + \sqrt{5} = 0$, find the minimum value of ${(a - c)}^2 + {(b - d)}^2$.

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