
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 prime number \( p > 3 \), we define the sequence \( \left\{a_{n}\right\} \) as follows: 
\[
a_{n} = 
\begin{cases} 
n & \text{when } n \in \{0, 1, \ldots, p-1\} \\
a_{n-1} + a_{n-p} & \text{when } n \geq p 
\end{cases}
\]
Find the remainder when \( a_{p^3} \) is divided by \( p \).

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