
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 $1$ and a common difference $d\ne 0$. Also, $\left( {{a}_{1}},{{a}_{2}},{{a}_{4}} \right)$ forms a geometric sequence.

(I) Find the general term formula for the sequence $\left\{ {{a}_{n}} \right\}$ and the sum of the first $n$ terms ${{S}_{n}}$;

(II) Let ${{b}_{n}}=\frac{1}{{{S}_{n}}}(n\in {{\mathbb{N}}^{*}})$, find the smallest positive integer $n$ that satisfies the inequality ${{b}_{1}}+{{b}_{2}}+\cdots +{{b}_{n}} > \frac{9}{5}$.

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