
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 the positive sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$, and for any $n \in \mathbb{N}^*$, it holds that $S_n = \left( \frac{a_n + 1}{2} \right)^2$.

$(1)$ Find the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n$;  
$(2)$ Let the sequence $b_n = a_n + \lambda$, $n \in \mathbb{N}^*$, $\lambda \in \mathbb{R}$, have a sum of the first $n$ terms denoted as $T_n$.  
$\quad$ $(i)$ If the minimum value of the sequence $\{T_n\}$ is $T_6$, find the range of the real number $\lambda$;  
$\quad$ $(ii)$ If for any two different terms in the sequence $\{b_n\}$, their sum is still an item in the sequence, then the sequence is called a "closed sequence". Is there such a "closed sequence" $\{b_n\}$, such that for any $n \in \mathbb{N}^*$, $T \neq 0$, and $\frac{1}{12} < \frac{1}{T_1} + \frac{1}{T_2} + \frac{1}{T_3} + \ldots + \frac{1}{T_n} < \frac{11}{18}$? If it exists, find all possible values of the real number $\lambda$; if not, please explain why.

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