
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 $\\{ {a_{n}} \\}$ that satisfies: ${a_{n+1}} > {a_{n}}\left( n\in {N^{*}} \right),{a_{1}}=1$, the first three terms of this sequence are added with $1$, $1$, $3$ respectively, and become the first three terms of a geometric sequence $\\{ {b_{n}} \\}$.
(I) Find the general term formulas for the sequences $\\{ {a_{n}} \\}$ and $\\{ {b_{n}} \\}$.
(II) Let ${{T_{n}}=\\frac{{{a_{1}}}{{{b_{1}}}}+\\frac{{{a_{2}}}{{{b_{2}}}}+\\cdot \\cdot \\cdot +\\frac{{{a_{n}}}{{{b_{n}}}}\left( n\in {N^{*}} \right),$ if ${{T_{n}}+\\frac{2n+3}{{{2^{n}}}}-\\frac{1}{n} < c\left( c\\in Z \right)}$ always holds true, find the minimum value of $c$.

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