
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
In the binomial expansion of \((\sqrt{x} - \frac{2}{x})^{12}\),
(I) find the coefficient of the term containing \(x^3\);
(II) if the binomial coefficients of the \(3k^{\text{th}}\) term and the \((k+2)^{\text{th}}\) term are equal, find the value of \(k\).

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