
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

a) How many distinct positive integer solutions exist for the equation

$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n ?
$$

b) How many non-negative integer solutions exist for the equation

$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n ?
$$

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