
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 a common difference $d\neq 0$, and the sum of its first $n$ terms is $S_{n}$. If $a_{2}+a_{8}=22$, and $a_{4}$, $a_{7}$, $a_{12}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If ${T}_{n}=\frac{1}{{S}_{1}}+\frac{1}{{S}_{2}}+…+\frac{1}{{S}_{n}}$, determine the range of values for $T_{n}$.

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