
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 $z_1=\sin x+\lambda i$ and $z_2=(\sin x+ \sqrt{3}\cos x)-i$ (where $\lambda, x\in\mathbb{R}$, and $i$ is the imaginary unit), solve the following:

(1) If $2z_1=i\cdot z_2$ and $x\in(0, \frac{\pi}{2})$, find the values of $x$ and $\lambda$.

(2) Assume the vectors corresponding to the complex numbers $z_1$ and $z_2$ on the complex plane are $\overrightarrow{OZ_{1}}$ and $\overrightarrow{OZ_{2}}$, respectively, and $\overrightarrow{OZ_{1}}\perp \overrightarrow{OZ_{2}}$. If $\lambda=f(x)$, determine the minimal positive period and the intervals of monotonic decrease for $f(x)$.

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