
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
Given the function $f(x) = 2a\cos^2{x} + b\sin{x}\cos{x} - \frac{\sqrt{3}}{2}$, and $f(0) = \frac{\sqrt{3}}{2}$, $f\left(\frac{\pi}{4}\right) = \frac{1}{2}$,
(1) Find the values of $a$ and $b$;
(2) Simplify the expression of function $f(x)$;
(3) Determine the range of $f(x)$.

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