
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 $a_{1}=1$, ${a}_{n+1}^{2}-a_{n+1}={a}_{n}^{2}+a_{n}$.  
$(1)$ 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 $S_{n}$.

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