
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
In the Cartesian coordinate system, $O$ is the origin, $\overrightarrow{OA}=(1,4)$, $\overrightarrow{OB}=(2,3)$, $\overrightarrow{OC}=(x,1)$.  
$(1)$ If points $A$, $B$, and $C$ are collinear, find the value of $x$.  
$(2)$ When $x=3$, does there exist a point $M$ on the line $OC$ such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ attains the minimum value? If yes, find the coordinates of point $M$; if not, explain the reason.

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