
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 sum of its first 5 terms equal to 50, and its 7th term ${a_{7}}=22$. Another sequence $\\{ {b_{n}} \\}$ has the sum of its first $n$ terms denoted as ${{S_{n}}}$, with ${{b_{1}}=1}$, and ${{b_{n+1}}=3{{S_{n}}+1}}$.

(1) Find the general term formulas for both sequences $\\{ {a_{n}} \\}$ and $\\{ {b_{n}} \\}$.

(2) If sequence $\\{ {c_{n}} \\}$ satisfies $\frac{{c_{1}}}{{{b_{1}}}}+\frac{{c_{2}}}{{{b_{2}}}}+\cdot \cdot \cdot \cdot +\frac{{c_{n}}}{{{b_{n}}}}={{a}_{{n+1}}}$, $n\in {{N}^{*}}$, find the value of ${{c}_{1}}+{{c}_{2}}+\cdot \cdot \cdot +{{c}_{{2017}}}$.

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