
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 $\left\{ a_n \right\}$ 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.

(Ⅰ) Find the general formula for the sequence $\left\{ a_n \right\}$;

(Ⅱ) Let $b_n=\frac{(a_n+1)^2}{a_na_{n+1}}$, $S_n=b_1+b_2+b_3+...+b_n$, is there a maximum integer $t$ such that for any $n \in \mathbb{N}^*$, $S_n > t$ always holds? If it exists, find the value of $t$; if not, explain why.

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