
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 \( x \) and \( y \) be non-negative real numbers that sum to 1. Compute the number of ordered pairs \((a, b)\) with \( a, b \in \{0, 1, 2, 3, 4\} \) such that the expression \( x^a y^b + y^a x^b \) has maximum value \( 2^{1-a-b} \).

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