
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 $\{a_n\}$ where the common difference $d > 0$, and the sum of the first $n$ terms is $S_n$, it is known that $a_2 \cdot a_3 = 45$ and $a_1 + a_5 = 18$.  
(1) Find the general formula for the $n$th term of the sequence $\{a_n\}$.  
(2) Let $b_n = \frac{S_n}{n + c}$ ($n \in \mathbb{N}^*$), is there a non-zero number $C$ that makes the sequence $\{b_n\}$ also an arithmetic sequence? If it exists, find the value of $c$; if not, explain why.

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