
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 that the sequence $\{a_n\}$ is an arithmetic sequence with a positive common difference, and its sum of the first $n$ terms is $S_n$, and $a_2 \cdot a_3 = 15$, $S_4 = 16$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Let the sequence $\{b_n\}$ satisfy $b_1 = a_1$, $b_{n+1} - b_n = \frac{1}{a_n \cdot a_{n+1}}$, find the general formula for the sequence $\{b_n\}$.

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