
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 $z\_1=\cos \alpha\_1+i\sin \alpha\_1$, $z\_2=\cos \alpha\_2+i\sin \alpha\_2$, $z\_3=\cos \alpha\_3+i\sin \alpha\_3$ (where $\alpha\_1$, $\alpha\_2$, $\alpha\_3\in R$), it is known that $z\_1\cdot z\_2=\cos (\alpha\_1+\alpha\_2)+i\sin (\alpha\_1+\alpha\_2)$ and $z\_2\cdot z\_3=\cos (\alpha\_2+\alpha\_3)+i\sin (\alpha\_2+\alpha\_3)$. Based on these conclusions, what is the value of $z\_1\cdot z\_2\cdot z\_3$?

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