
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 of parabola $C$ is at the origin $(0,0)$ and its focus is $F(0,1)$,  
(1) Find the equation of parabola $C$;  
(2) Draw line $l$ passing through point $F$ intersecting parabola $C$ at points $A$ and $B$. If lines $AO$ and $BO$ intersect line $y=x-2$ at points $M$ and $N$ respectively, determine the range of values for $|MN|$.

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