
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 in the sequence \(\{a_n\}\), the consecutive terms \(a_n, a_{n+1}\) are the two real roots of the equation
\[ x^2 - nx + c_n = 0 \quad (n=1,2,\ldots) \]
and \(a_1 = 1\),

(1) Find the general term formula of the sequence \(\{a_n\}\);  
(2) Let \(b_n = c_{2n-1}\). Find the general term formula of the sequence \(\{b_n\}\) and the sum \(T_n\) of the first \(n\) terms of \(\{b_n\}\).

(If necessary, you may use the following formula:
\[
1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}.
\])

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