
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 $z\_1=2\sin θ- \sqrt {3}i$ and $z\_2=1+(2\cos θ)i$, where $i$ is the imaginary unit and $θ∈[\frac{π}{3}, \frac{π}{2}]$.
1. If $z\_1⋅z\_2$ is a real number, find the value of $\cos 2θ$.
2. If the complex numbers $z\_1$ and $z\_2$ correspond to vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively, find the range of real number $λ$ that satisfies the equation $(λ \overrightarrow{a}- \overrightarrow{b})⋅( \overrightarrow{a}-λ \overrightarrow{b})=0$.

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