
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 $S_n$, and $a_1=1$, $a_{n+1}=2S_n+1$ for $n\in \mathbb{N}^*$. In the arithmetic sequence $\{b_n\}$, $b_2=5$, and the common difference $d=2$.

$(I)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

$(II)$ Determine whether there exists a positive integer $n$ such that $a_1b_1+a_2b_2+\cdots+a_nb_n > 60n$. If it exists, find the minimum value of $n$.

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