
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 that the expansion of $(1+x+x^2+\cdots+x^{p-1})^{p+1}$ is $a_0 + a_1 x + a_2 x^2 + \cdots + a_{p^2-1} x^{p^2-1}$, where $p$ is an integer not less than 3, find the value of $a_0 + a_p + a_{2p} + \cdots + a_{p^2-p}$.

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