
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 that $\lambda \in \mathbb{R}$, $f(x) = \vec{a} \cdot \vec{b}$, where $\vec{a} = (\cos x, \sin x)$, $\vec{b} = (\lambda \sin x - \cos x, \cos (\frac{\pi}{2} - x))$, and it is known that $f(x)$ satisfies $f(-\frac{\pi}{3}) = f(0)$.
1. Find the interval where the function $f(x)$ is strictly increasing.
2. Solve the inequality $2\cos(2x - \frac{\pi}{6}) > \sqrt{3}$.

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