
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 vertex $B(5,1)$ of $\triangle ABC$ and the equation of the altitude on side $AB$ is $x-2y-5=0$.

$(Ⅰ)$ Find the equation of line $AB$.

$(Ⅱ)$ Choose one of the following conditions and find the equation of line $AC$.
1. The equation of the angle bisector of angle $A$ is $x+2y-13=0$.
2. The equation of the median on side $BC$ is $2x-y-5=0$.

Note: If multiple conditions are chosen and solved separately, the first solution will be scored.

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