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

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