
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
(1) Find all natural numbers \( n \) for which \( S_{n} = 9 + 17 + 25 + \cdots + (8n + 1) \) is a perfect square.

(2) Find all natural numbers \( n \) for which \( T_{n} = 5 + 11 + 17 + \cdots + (6n - 1) \) is a perfect square.

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