
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 an arithmetic sequence $\{a_n\}$ with a common difference $d > 0$, its sum of the first $n$ terms is $S_n$, and it satisfies: $a_2 \cdot a_3 = 45$, $a_1 + a_4 = 14$.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Construct a new sequence $\{b_n\}$ using the formula $b_n = \frac{S_n}{n + c}$. If $\{b_n\}$ is also an arithmetic sequence, find the non-zero constant $c$;
(III) Find the maximum value of $f(n) = \frac{b_n}{(n + 25) \cdot b_{n + 1}}$, where $n \in \mathbb{N}^*$.

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