
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 $d > 0$, let $S_n$ denote the sum of its first $n$ terms. If $a_2 \cdot a_3 = 45$ and $a_1 + a_5 = 18$, find:
1. The general term formula for the sequence $\{a_n\}$.
2. Determine if there exists a non-zero number $C$ such that the sequence $\{b_n\} = \frac{S_n}{n+C}$ is also an arithmetic sequence. If such a number exists, find its value; otherwise, explain why not.

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