
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 $A(1,1)$, the equation of the line containing the altitude $CD$ is $3x+y-12=0$, and the equation of the angle bisector of angle $B$, $BE$, is $x-2y+4=0$. Find:  
1. The coordinates of point $B$;  
2. The equation of line $BC$.

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