
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 plane rectangular coordinate system $xOy$, the three vertices of $\triangle ABC$ are $A(-2,0)$, $B(0,4)$, $C(m,n)$, where point $C$ lies on the line $x-3y-3=0$. <br/>$(1)$ If $m=3$, find the equation of the median on side $AB$ of $\triangle ABC$; <br/>$(2)$ If $\triangle ABC$ is a right triangle, find the value of the real number $m$.

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