
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 that the vertices of triangle $\triangle ABC$ are $A(0,4)$, $B(-2,6)$, and $C(-8,0)$, find:<br/>
$(1)$ The equation of the median on side $AC$;<br/>
$(2)$ The equation of the altitude on side $AC$;<br/>
$(3)$ The equation of the perpendicular bisector on side $AC$.

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