
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 $f(x)=2a\cos ^{2}x+b\sin x\cos x$, with $f(0)=2$, and $f(\frac{\pi}{3})=\frac{1+\sqrt{3}}{2}$.

1. Find the maximum and minimum values of $f(x)$.
2. Find the interval(s) where $f(x)$ is strictly increasing.
3. For angles $\alpha$ and $\beta$, if $\alpha-\beta\neq k\pi$, $k\in\mathbb{Z}$, and $f(\alpha)=f(\beta)$, find the value of $\tan(\alpha+\beta)$.

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