
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 curve $C$ is above the $x$-axis, and the distance from any point on curve $C$ to point $F(0,1)$ is smaller than the distance to line $y=-2$ by $1$.

(I) Find the equation of curve $C$;

(II) Let $m > 0$, a line passing through point $M(0,m)$ intersects curve $C$ at points $A$ and $B$.
  
  1. If $\triangle AFB$ is an equilateral triangle, find the value of real number $m$;
  2. If $\overrightarrow{FA} \cdot \overrightarrow{FB} < 0$, find the range of real number $m$.

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