
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
1. Find the number of positive integer solutions to the equation \( x_{1} + x_{2} + x_{3} + \cdots + x_{m} = n \) where \( m \) and \( n \) are natural numbers and \( m < n \).

2. Find the number of integer solutions to the equation \( x_{1} + x_{2} + x_{3} + \cdots + x_{m} = n \) where \( m \) and \( n \) are natural numbers, \( n \geq (m-2)r + 1 \) and \( r \) is an integer, under the conditions \( x_{1} \geq 1 \), \( x_{i} \geq r \) for \( i = 2, 3, \ldots, m-1 \), and \( x_{m} \geq 0 \).

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