
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 sequence $\{a_n\}$ that satisfies $2a_{n+1}=a_n$ and $a_1= \frac{1}{2}$.

$(1)$ Find $a_n$;  
$(2)$ If $b_n=\log_{\frac{1}{2}}a_n$, find the sum of the first $n$ terms, $T_n$, of the sequence $\left\{ \frac{1}{b_nb_{n+1}}\right\}$.

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