
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 $a_1=1$ and common difference $d > 0$, and the second, fifth, and fourteenth terms are respectively the second, third, and fourth terms of a geometric sequence,

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

(2) Let $b_n=\dfrac{(a_n+1)^2}{a_n \cdot a_{n+1}}$, $S_n=b_1+b_2+\ldots+b_n$, determine if there exists a maximum integer $t$ such that for any $n\in\mathbb{N}^*$, $S_n > t$ always holds. If such a $t$ exists, find its value; if not, explain why.

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