
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]


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Given the functions $f(x) = \sin x$ and $g(x) = f(x) - ax$, where $x \in [0, \frac{\pi}{2}]$.

(1) When $a = \frac{1}{2}$, find the interval of monotonic increase for the function $g(x)$.

(2) If the minimum value of the function $g(x)$ is 0, find the range of the real number $a$.

(3) Let $0 \leq x_1 < x_2 \leq \frac{\pi}{2}$, compare the magnitude of $- \frac{f(x_1) + f(x_2)}{2}$ and $\frac{f'(x_2) - f'(x_1)}{x_2 - x_1}$, and explain the reason.

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