
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 $\triangle ABC$, the equation of the line containing the altitude on side $BC$ is $x-2y-1=0$, and the equation of the angle bisector of angle $A$ is $y=0$. If the coordinates of point $B$ are $\left(2,1\right)$.<br/>$(1)$ Find the coordinates of points $A$ and $C:<br/>$(2)$ Find the equation of the line $l$ containing the altitude on side $AC$.

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