
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 a triangle $\triangle ABC$ with vertices $A(4, 0)$, $B(8, 10)$, and $C(0, 6)$.
(1) Find the equation of the line on which the altitude from vertex $A$ to side $BC$ lies.
(2) Find the equation of the line passing through point $B$ such that the distances from points $A$ and $C$ to this line are equal.

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