
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 $\triangle ABC$, point $A(2,-1)$, the equation of the median on side $AB$ is $x+3y-6=0$, and the equation of the internal angle bisector of $\angle ABC$ is $x-y+1=0$.  
$(1)$ Find the coordinates of point $B$.  
$(2)$ Find the equation of the line containing side $BC$ of triangle $\triangle ABC$.

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