
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 that $S_{5}=50$, $S_{1}$, $S_{2}$, $S_{4}$ form a geometric sequence, and $S_{6}=3(a_{6}+2)$, choose any two of these three conditions and add them to the following question, then solve the problem.<br/>Question: For an arithmetic sequence $\{a_{n}\}$ with a common difference of $d\left(d\neq 0\right)$, where the sum of the first $n$ terms is $S_{n}$, and it satisfies _____.<br/>$(1)$ Find $a_{n}$;<br/>$(2)$ If $b_{n}-b_{n-1}=2a_{n}(n\geqslant 2)$, and $b_{1}-a_{1}=1$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{{{b_n}}}\}$ denoted as $T_{n}$.

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