
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 $\triangle ABC$, given $A(1,1)$ and $B(3,-2)$.  
$(1)$ If the line $l$ passes through point $M(2,0)$ and the distances from points $A$ and $B$ to $l$ are equal, find the equation of line $l$.  
$(2)$ If the angle bisector of $\angle C$ lies on the line with equation $x+y-3=0$, find the equation of line $BC$.

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