
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

Assume that \( f(x, y) \) is defined for all positive integers \( x \) and \( y \), and that the following equations are satisfied:
$$
\begin{aligned}
f(0, y) & = y + 1, \\
f(x + 1, 0) & = f(x, 1), \\
f(x + 1, y + 1) & = f(x, f(x + 1, y)).
\end{aligned}
$$

Determine \( f(2, 2) \), \( f(3, 3) \), and \( f(4, 4) \). 
Alternative version: Determine \( f(4, 1981) \).

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