
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 with positive terms $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and it is known that $2a_2=S_2+ \frac {1}{2},a_3=2$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ If $b_n=\log_2a_n+3$, and the sum of the first $n$ terms of the sequence $\left\{ \frac {1}{b_nb_{n+1}}\right\}$ is $T_n$, find the minimum positive integer $n$ that satisfies $T_n > \frac {1}{3}$.

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