
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 $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $n\in\mathbb{N}^*$.  
$(1)$ Find $a_n$;  
$(2)$ If the sequence $\{b_n\}$ satisfies $b_n= \frac{1}{(1+\log_{3}a_n)(3+\log_{3}a_n)}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, and for any positive integer $n$, $T_n < m$, find the minimum value of $m$.

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