
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
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, where $a_{n} \gt 0$, and $a_n^2+2{a_n}+1=4{S_n}$. <br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{{{{({-1})}^n}({\frac{{4n}}{{{a_n}{a_{n+1}}}}})}\right\}$.

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