
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 $\{a_n\}$ satisfying $a_1 = a$ ($a \in \mathbb{N}^*$), and $a_1 + a_2 + \ldots + a_n - pa_{n+1} = 0$ ($p \neq 0$, $p \neq -1$, $n \in \mathbb{N}^*$).  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) For every positive integer $k$, if $a_{k+1}$, $a_{k+2}$, $a_{k+3}$ are arranged in ascending order and can form an arithmetic sequence with common difference $d_k$, find the value of $p$ and the corresponding sequence $\{d_k\}$.

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