
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 $f(x) = \cos x \sin(x + \frac{\pi}{3}) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}$,
1. Find the intervals where $f(x)$ is monotonically increasing.
2. Let $g(x) = 2af(x) + b$. If the range of $g(x)$ on $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ is $[2, 4]$, find the values of $a$ and $b$.

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