
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 $\left| a_{n+1} - a_n \right| = p^n$ for $n\in\mathbb{N}^*$.  
(1) If $p=1$, list all possible values of $a_4$.  
(2) If the sequence $\{a_n\}$ is increasing, and $a_1, 2a_2, 3a_3$ form an arithmetic sequence, find the value of $p$.  
(3) If $p= \frac{1}{2}$, and $\{a_{2n-1}\}$ is an increasing sequence while $\{a_{2n}\}$ is a decreasing sequence, find the general term formula for the sequence $\{a_n\}$.

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