
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 triangle $\triangle ABC$, the coordinates of point $C$ are $(4,-1)$, and the equation of the median line on side $BC$ is $3x-y-1=0$. The slope of line $AC$ is $\frac{3\pi}{4}$.  
$(1)$ Find the coordinates of point $A$.  
$(2)$ The line $l$ passing through point $A$ intersects the positive half-axis of the $x$-axis and the positive half-axis of the $y$-axis at points $M$ and $N$, respectively. Find the minimum value of the area of $\triangle MON$ (where $O$ is the origin).

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