
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 that the sequence $\{a_n\}$ is an arithmetic sequence with the sum of the first $n$ terms denoted as $S_n$, and $a_2=4$, $S_5=30$.

(1) Find $a_n$;

(2) Let $b_n=a_n\cdot 2^{a_n}$ for $n\in\mathbb{N^*}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $B_n$;

(3) Let $c_n=\frac{1}{a_n-1}$ for $n\in\mathbb{N^*}$, there exists a positive real number $k$ such that $(1+c_1)(1+c_2)\cdots(1+c_n)\geq k\sqrt{2n+1}$ holds for all $n\in\mathbb{N^*}$, find the maximum value of $k$.

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