
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that are not collinear, and $t$ is a real number.
(I) If $\overrightarrow{OA} = \overrightarrow{a}$, $\overrightarrow{OB} = t\overrightarrow{b}$, and $\overrightarrow{OC} = \frac{1}{3}(\overrightarrow{a} + \overrightarrow{b})$, find the value of $t$ when points $A$, $B$, and $C$ are collinear.
(II) If $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, find the range of values for $|\overrightarrow{a} - x\overrightarrow{b}|$ where $x \in [-1, \frac{1}{2}]$.

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