
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 complex numbers $z_1 = \sin(2x) + \lambda i$ and $z_2 = m + (m - \sqrt{3} \cos(2x))i$, where $\lambda, m, x \in \mathbb{R}$, and $z_1 = z_2$.
1. If $\lambda = 0$ and $0 < x < \pi$, find the value of $x$.
2. Let $\lambda = f(x)$, and it is known that when $x = \alpha$, $\lambda = \frac{1}{2}$. Find the value of $\cos(4\alpha + \frac{\pi}{3})$.

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