
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 the triangle $\triangle ABC$ with vertex A at (5,1), the equation of the bisector of $\angle B$ is $x+y-5=0$, and the equation of the median from vertex A to side $AB$, $CM$, is $2x-y-5=0$. Find:
(1) The coordinates of vertex B;
(2) The equation of line BC.

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