
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
The sequence \(\{a_n\}\) satisfies \(a_1 = 3\), and for any positive integers \(m, n\), \(a_{m+n} = a_{m} + a_{n} + 2mn\).
(1) Find the general term formula of \(\{a_n\}\);
(2) If there exists a real number \(c\) such that \(\sum_{i=1}^{k} \frac{1}{a_i} < c\) holds for all positive integers \(k\), determine the range of values for \(c\).

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