
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 $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, $a_1=1$, and $a_{n+1}=2S_n+1$ for $n\in\mathbb{N}^*$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) Let $c=\log_{3}a_{2n}$, $b_n= \frac{1}{c_n\cdot c_{n+2}}$, and denote the sum of the first $n$ terms of the sequence $\{b_n\}$ as $T_n$. If for any $n\in\mathbb{N}^*$, $\lambda < T_n$ always holds, find the range of the real number $\lambda$.

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