
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 $\{a_n\}$ that satisfies $a_1=1$ and $|a_{n+1}-a_n|=p^n$ for $n\in\mathbb{N}^*$, let $S_n$ denote the sum of the first $n$ terms of the sequence $\{a_n\}$.

(1) If $\{a_n\}$ is an increasing sequence, and $a_1$, $2a_2$, $3a_3$ form an arithmetic sequence, find the value of $p$.

(2) If $p=\frac{1}{2}$, and the subsequence $\{a_{2n-1}\}$ is increasing while the subsequence $\{a_{2n}\}$ is decreasing, find a general formula for the sequence $\{a_n\}$.

(3) If $p=1$, for a given positive integer $n$, determine if there exists a sequence $\{a_n\}$ satisfying the given conditions such that $S_n=n$. If such a sequence exists, provide an example. If not, explain why.

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