
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 circumference of one base of a frustum is three times that of the other base, and the slant height is 3. If the lateral area of the frustum is $84\pi$, then the radius of the smaller base is ______.

(2) The Provincial Hospital plans to arrange for 6 doctors to be on duty from May 1st to May 3rd, requiring each person to be on duty for 1 day, with 2 people arranged per day. If among the 6 doctors, Doctor A cannot be on duty on the 2nd, and Doctor B cannot be on duty on the 3rd, then the total number of different duty arrangements is ______.

(3) Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$. If $a_1=3$ and $S_n=\frac{1}{2}a_{n+1}+1$, then the general formula for $\{a_n\}$ is ______.

(4) Let the function $f(x)$ have a domain $D$. If the function $y=f(x)$ satisfies the following two conditions, it is called a closed function on the domain $D$: ① $y=f(x)$ is a monotonic function on $D$; ② There exists an interval $[a,b]\subseteq D$ such that the range of $f(x)$ on $[a,b]$ is $[a,b]$. If the function $f(x)=\sqrt{2x+1}+k$ is a closed function, then the range of values for $k$ is ______.

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