
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]


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Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear non-zero vectors $(t\in\mathbb{R})$  
$(1)$ Let $\overrightarrow{OA}= \overrightarrow{a}$, $\overrightarrow{OB}=t \overrightarrow{b}$, $\overrightarrow{OC}= \dfrac {1}{3}( \overrightarrow{a}+ \overrightarrow{b})$, then for what value of the real number $t$ are the points $A$, $B$, and $C$ collinear?  
$(2)$ If $| \overrightarrow{a}|=| \overrightarrow{b}|=1$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120 ^{\circ}$, then for what value of the real number $x$ is the value of $| \overrightarrow{a}-x \overrightarrow{b}|$ minimized?

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