
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 set of positive integers \( p_{1}, p_{2}, \cdots, p_{n} \) and an arbitrary positive integer \( r \), find the number of non-negative integer solutions \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) to the equation \( p_{1} x_{1} + p_{2} x_{2} + \cdots + p_{n} x_{n} = r \).

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