
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
In the geometric sequence $\{a_n\}$, $a_4= \frac{2}{3}$, $a_3+a_5= \frac{20}{9}$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ If the common ratio of the sequence $\{a_n\}$ is greater than $1$, and $b_n=\log_3 \frac{a_n}{2}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$.

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