
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
Find the largest integer pair \( (a, b) \) such that
\[ \sin^a x \cdot \cos^b x \geq \left(\frac{1}{2}\right)^{(a+b)/2} \]
for all \( x \) in the range \( 0 \leq x \leq \frac{\pi}{2} \).

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