
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 geometric sequence $\{a_n\}$ with the first term equal to $\frac{1}{16}$ and a common ratio not equal to $1$, let $S_n$ be the sum of its first $n$ terms, satisfying $S_3=4S_2- \frac{5}{16}$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Let $b_n=\log_a a_n (a > 0$ and $a \neq 1)$, find the maximum and minimum values of the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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