
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 person has $n$ red, $n$ green, and $n$ black matches in a box. Each time, they randomly take one match. When all the black matches are used up, there are $r$ red matches and $g$ green matches left. Find the number of different ways this outcome can occur.

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