
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 center of circle $C$ lies on the line $x-2y=0$ and is tangent to the $y$-axis at the point $(0,1)$.  
$(Ⅰ)$ Find the equation of circle $C$;  
$(Ⅱ)$ If circle $C$ intersects the line $l:x-y+m=0$ at points $A$ and $B$, _____, find the value of $m$.  
Choose one of the following conditions to add to the above question and answer:  
Condition ①: $\angle ACB=120^{\circ}$;  
Condition ②: $|AB|=2\sqrt{3}$.

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