
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 complex numbers $z\_1=\cos x+i$ and $z\_2=1-i\sin x$, where $x\in\mathbb{R}$:

1. Find the minimum value of $|z\_1-z\_2|$.
2. Let $z=z\_1\cdot z\_2$, and denote by $f(x)=\operatorname{Im}(z)$ the imaginary part of $z$. By stretching the horizontal coordinates of the graph of $f(x)$ by a factor of 2 (while keeping the vertical coordinates unchanged) and then shifting the resulting graph to the right by $\frac{\pi}{2}$ units, obtain the analytical expression for the new function $g(x)$.

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