
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 $\{a_n\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n = \frac{1}{\log_{2}a_n \cdot \log_{2}a_{n+1}}$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the minimum value of $T_n$.

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