
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 a sequence $\{a_{n}\}$ where all terms are positive, and ${a_1}=2,a_{n+1}^2-2{a_{n+1}}=a_n^2+2{a_n}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{{\frac{1}{{{a_n}{a_{n+2}}}}}\right\}$ be denoted as $S_{n}$, find the range of values for $S_{n}$.

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