
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 that $k$ and $n$ are positive integers such that $n = 3k$, evaluate the expression $\left(\frac{n - 3k - 2}{k + 2}\right)C^n_k$ where $C^n_k = \frac{n!}{k! (n-k)!}$.
**A) Always an integer**
**B) Never an integer**
**C) An integer when k = 0 or k = 2**
**D) An integer for some specific values not described**

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