
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 planar vectors $\overrightarrow{α}$ and $\overrightarrow{β}$ that satisfy $|2\overrightarrow{β} - \overrightarrow{α}| = \sqrt{3}$, and the angle between $\overrightarrow{α} + \overrightarrow{β}$ and $\overrightarrow{α} - 2\overrightarrow{β}$ is $150^{\circ}$, find the minimum value of $|t(\overrightarrow{α} + \overrightarrow{β}) - \frac{3}{2}\overrightarrow{β}|$ $(t \in \mathbb{R})$.

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