
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 $a_{1}=1$, and ${a_{n+1}}=\frac{{a_n}}{{2+a_n}}$ for $n\in\mathbb{N^*}$, which of the following conclusions are correct?<br/>① $\left\{\frac{1}{{a_n}}+1\right\}$ forms a geometric sequence;<br/>② The general formula for $\{a_{n}\}$ is ${a_n}=\frac{1}{{2^n}-1}$;<br/>③ $\{a_{n}\}$ is a decreasing sequence;<br/>④ The sum of the first $n$ terms of $\{a_{n}\}$, denoted as $T_{n}$, is less than $2$.

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