
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
Let \( T_k \) denote the product of the first \( k \) odd prime numbers: \( T_1 = 3, T_2 = 3 \cdot 5, T_6 = 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \), and so on. For each natural number \( k \), find the number of natural numbers \( n \) such that the number \( n^2 + T_k n \) is a perfect square of a natural number. Solve the problem for: 
a) \( k = 1 \), 
b) \( k = 2 \), 
c) an arbitrary given natural \( k \).

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