
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$ and $4S_n = a_n \cdot a_{n+1}, n \in \mathbb{N}^*$.  
Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n^2} \right\}$ be $T_n$, find the range of $T_n$ such that $\frac{n}{4n+4} < T_n < \frac{1}{2}$.

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