
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 in triangle $\triangle ABC$, the equation of the line containing the height from side $BC$ is $x-2y+1=0$, the equation of the angle bisector of $\angle A$ is $y=0$, and the coordinates of point $C$ are $(1,2)$.
(Ⅰ) Find the coordinates of points $A$ and $B$;
(Ⅱ) A line $l$ passes through point $C$ and intersects the positive half-axes of the x-axis and y-axis at points $M$ and $N$, respectively. Find the minimum area of $\triangle MON$ and the equation of line $l$ at this time.

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