
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
Let \( R \) be the reals. Define \( f : [0, 1) \times [0, 1) \to R \) by \( f(x, y) = \sum x^m y^n \), where the sum is taken over all pairs of positive integers \( (m, n) \) satisfying \( m \ge n/2 \), \( n \ge m/2 \). Find \( \lim_{{(x, y) \to (1, 1)}} (1 - xy^2)(1 - x^2 y) f(x, y) \).

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