
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
Find a positive integer \( k \), such that
(a) For any positive integer \( n \), there does not exist \( j \) that satisfies \( 0 \leqslant j \leqslant n-k+1 \), and \( C_{n}^{j}, C_{n}^{j+1}, \cdots, C_{n}^{j+k-1} \) form an arithmetic sequence;
(b) There exists a positive integer \( n \), such that there is \( j \) that satisfies \( 0 \leqslant j \leqslant n-k+2 \), and \( C_{n}^{j}, C_{n}^{j+1}, \cdots, C_{n}^{j+k-2} \) form an arithmetic sequence.
Further, determine all \( n \) having property (b).

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