
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 positive sequence $\left\{ a_n \right\}$ with the sum of the first $n$ terms denoted as $S_n$, and $4S_n = (a_n + 1)^2$.

(Ⅰ) Find the general formula for the sequence $\left\{ a_n \right\}$;

(Ⅱ) Find the sum of the first $n$ terms, $T_n$, for the sequence $\left\{ \frac{2}{a_n a_{n+1}} \right\}$;

(Ⅲ) If for any $n \in N^*$, $T_n \geqslant k$ always holds, find the range of values for $k$.

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