
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
In the arithmetic sequence $\{a_{n}\}$, where $a_{1}=1$ and $S_{n}$ is the sum of the first $n$ terms of $\{a_{n}\}$, and $\left\{{\sqrt{{S_n}}}\right\}$ is also an arithmetic sequence.  
$(1)$ Find $a_{n}$;  
$(2)$ Let ${b_n}=\frac{{{S_n}}}{{{a_n}{a_{n+1}}} ({n∈{N^*}})$. Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{n}$.

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