
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 the first term $a_{1}=1$, common difference $d > 0$, and the second, fifth, and fourteenth terms forming a geometric sequence in that order.

1. Find the general term formula for the sequence $\{a_{n}\}$.
2. Define $b_{n}= \frac{1}{n(a_{n}+3)} (n∈\mathbb{N}^{*})$, $S_{n}=b_{1}+b_{2}+…+b_{n}$. Determine if there exists a $t$ such that $S_{n} > \frac{t}{36}$ holds for any $n$. If it exists, find the maximum integer value of $t$; if not, explain the reason.

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