
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
The function \( f: \mathbb{Z}^{2} \rightarrow \mathbb{Z} \) satisfies the following conditions:
- \( f(x, 0) = f(0, y) = 0 \)
- \( f(x, y) = f(x-1, y) + f(x, y-1) + x + y \)

for all nonnegative integers \( x \) and \( y \). Find \( f(6, 12) \).

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