
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
1. The maximum value of the function $f(x)=x(1-x)$, where $x\in(0,1)$, is _______.

2. In the sequence $\{a_n\}$, the sum of the first $n$ terms $S_n=3\cdot2^n+k$. If the sequence $\{a_n\}$ is a geometric sequence, then the value of the constant $k$ is _______.

3. In $\triangle ABC$, where $A=60^{\circ}$, $b=1$, and the area is $\sqrt{3}$, then $\frac{a+b+c}{\sin A+\sin B+\sin C}=$ _______.

4. Given the sequence $\{a_n\}$ satisfies $3a_1+2^2a_2+2^3a_3+\cdots+2^na_n=4^n-1$, the general term formula of $\{a_n\}$ is _______.

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