
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 that $θ$ is the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}| = 2$ and $|\overrightarrow{b}| = 1$, the quadratic equation concerning $x$: $x^{2} - |\overrightarrow{a}|x + \overrightarrow{a} \cdot \overrightarrow{b} = 0$ has real roots.

1. Find the range of $θ$.
2. Under the conditions of (1), find the maximum and minimum values of the function $f(θ) = \sin\left(2θ + \frac{π}{3}\right)$ and the corresponding $θ$ values.

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