
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 the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, and $S_n = \frac{1}{2}n^2 + \frac{11}{2}n (n \in \mathbf{N}^*)$.

(1) Find the general term formula of the sequence $\{a_n\}$;

(2) Let $c_n = \frac{1}{(2a_n - 11)(2a_n - 9)}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$. Find the maximum positive integer $k$ such that the inequality $T_n > \frac{k}{2014}$ holds for all $n \in \mathbf{N}^*$;

(3) Let $f(n) = \begin{cases} a_n (n = 2k - 1, k \in \mathbf{N}^*) \\ 3a_n - 13 (n = 2k, k \in \mathbf{N}^*) \end{cases}$. Does there exist $m \in \mathbf{N}^*$ such that $f(m + 15) = 5f(m)$ holds? If so, find the value of $m$; if not, explain the reason.

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