
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 $a$ and $b$ be two positive integers such that $a + b$ is even. Which of the following is the correct relationship between the arithmetic mean $\frac{a+b}{2}$ and their geometric mean $\sqrt{ab}$?
A) $\frac{a+b}{2} > \sqrt{ab}$
B) $\frac{a+b}{2} < \sqrt{ab}$
C) $\frac{a+b}{2} = \sqrt{ab}$ (only if $a=b$)
D) $\frac{a+b}{2} \leq \sqrt{ab}$
E) $\frac{a+b}{2} \geq \sqrt{ab}$

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