
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 $\{a_n\}$, for any $n \in \mathbb{N}^*$, it satisfies the equation $(kn + b)(a_1 + a_n) + p = 2(a_1 + a_2 + \dots + a_n)$, where $k$, $b$, and $p$ are constants.
1. For $k=0$, $b=3$, $p=-4$, find the sum $a_1+a_2+a_3+\dots+a_n$;
2. For $k=1$, $b=0$, $p=0$, if $a_3=3$ and $a_9=15$, find the general term formula for the sequence $\{a_n\}$;
3. A sequence $\{a_n\}$ is termed "closed" if the sum of any two (distinct) terms is also an element of the sequence. For $k=1$, $b=0$, $p=0$, and given $a_2 - a_1 = 2$, determine whether there exists such a "closed" sequence $\{a_n\}$ such that for any $n \in \mathbb{N}^*$, $S_n \neq 0$, and $$\frac{1}{12} < \frac{1}{S_1} + \frac{1}{S_2} + \frac{1}{S_3} + \dots + \frac{1}{S_n} < \frac{11}{18}.$$ If it exists, find all possible values for the first term $a_1$; if not, explain why.

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