
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 $S_n$ be the sum of the first $n$ terms of $\{a_n\}$, with $a_1=1$ and $S_2 \cdot S_3=36$.
- (I) Find $d$ and $S_n$;
- (II) Find the values of $m$ and $k$ ($m,k \in \mathbb{N}^*$) such that $a_m + a_{m+1} + a_{m+2} + \ldots + a_{m+k} = 65$.

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