
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 full score of this question is 12 points) Suppose the sequence $\{a_n\}$, where each term is positive, has a sum of the first $n$ terms denoted as $S_n$. It is known that $2a_2 = a_1 + a_3$, and the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $d$.
   (I) Find the general formula for the sequence $\{a_n\}$ (expressed in terms of $n$ and $d$);
   (II) Let $c$ be a real number. For any positive integers $m$, $n$, and $k$ that satisfy $m + n = 3k$ and $m \neq n$, the inequality $S_m + S_n > cS_k$ always holds. Find the maximum value of $c$.

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